# Finite-Support Polynomial Operations, Zero-Divisor Degree Collapse and an Infinite Carrier

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 3, 0, 0]. Give (raw,zero,normalized,degree).

View answer and explanation([1, 3, 0, 0], 0, [1, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 0, 0], 0, [1, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0001/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 2. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 0, 0, 0]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 0, 0, 0], 0, [3], 0)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 0, 0, 0], 0, [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0002/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 3. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 0, 0]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 2, 0, 0], 0, [3, 2], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 0, 0], 0, [3, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0003/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 4. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[0, 3, 0, 0]. Give (raw,zero,normalized,degree).

View answer and explanation([0, 3, 0, 0], 0, [0, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([0, 3, 0, 0], 0, [0, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0004/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 5. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 0, 0]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 2, 0, 0], 0, [3, 2], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 0, 0], 0, [3, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0005/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 6. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 0, 0, 0]. Give (raw,zero,normalized,degree).

View answer and explanation([2, 0, 0, 0], 0, [2], 0)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 0, 0, 0], 0, [2], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0006/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 7. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 0, 1, 1]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 0, 1, 1], 1, [3, 0], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 0, 1, 1], 1, [3, 0], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0007/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 8. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 3, 1, 1]. Give (raw,zero,normalized,degree).

View answer and explanation([1, 3, 1, 1], 1, [1, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 1, 1], 1, [1, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0008/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 9. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 1, 1]. Give (raw,zero,normalized,degree).

View answer and explanation([2, 3, 1, 1], 1, [2, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 1, 1], 1, [2, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0009/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 10. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 1, 1]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 1, 1, 1], 1, [3], 0)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 1, 1], 1, [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0010/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 11. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 1, 1]. Give (raw,zero,normalized,degree).

View answer and explanation([2, 3, 1, 1], 1, [2, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 1, 1], 1, [2, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0011/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 12. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 1, 1]. Give (raw,zero,normalized,degree).

View answer and explanation([1, 2, 1, 1], 1, [1, 2], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 1, 1], 1, [1, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0012/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 13. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[0, 3, 2, 2]. Give (raw,zero,normalized,degree).

View answer and explanation([0, 3, 2, 2], 2, [0, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([0, 3, 2, 2], 2, [0, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0013/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 14. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 2, 2]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 2, 2, 2], 2, [3], 0)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 2, 2], 2, [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0014/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 15. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 2, 2]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 1, 2, 2], 2, [3, 1], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 2, 2], 2, [3, 1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0015/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 16. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 2, 2]. Give (raw,zero,normalized,degree).

View answer and explanation([2, 3, 2, 2], 2, [2, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 2, 2], 2, [2, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0016/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 17. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 2, 2]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 1, 2, 2], 2, [3, 1], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 2, 2], 2, [3, 1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0017/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 18. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 2, 2]. Give (raw,zero,normalized,degree).

View answer and explanation([1, 2, 2, 2], 2, [1], 0)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 2, 2], 2, [1], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0018/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 19. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 0, 3, 3]. Give (raw,zero,normalized,degree).

View answer and explanation([2, 0, 3, 3], 3, [2, 0], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 0, 3, 3], 3, [2, 0], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0019/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 20. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 2, 3, 3]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 2, 3, 3], 3, [3, 2], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 2, 3, 3], 3, [3, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0020/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 21. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 3, 3]. Give (raw,zero,normalized,degree).

View answer and explanation([1, 2, 3, 3], 3, [1, 2], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 3, 3], 3, [1, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0021/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 22. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[2, 3, 3, 3]. Give (raw,zero,normalized,degree).

View answer and explanation([2, 3, 3, 3], 3, [2], 0)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([2, 3, 3, 3], 3, [2], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0022/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 23. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 2, 3, 3]. Give (raw,zero,normalized,degree).

View answer and explanation([1, 2, 3, 3], 3, [1, 2], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 2, 3, 3], 3, [1, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0023/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 24. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[3, 1, 3, 3]. Give (raw,zero,normalized,degree).

View answer and explanation([3, 1, 3, 3], 3, [3, 1], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([3, 1, 3, 3], 3, [3, 1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0024/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 25. Normalize a Finite-Support Coefficient Array

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Original coefficients=[1, 3, 0, 0, 0]. Give (raw,zero,normalized,degree).

View answer and explanation([1, 3, 0, 0, 0], 0, [1, 3], 1)

Retain the raw input array in the first field, then remove trailing actual-zero labels. Empty input and all-zero input normalize to [zero]. The exact result, retaining the requested order, is ([1, 3, 0, 0, 0], 0, [1, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0025/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-normalize · candidate





## 26. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0], right=[2, 1, 0, 0]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([1, 3], [2, 1], [3], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [2, 1], [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0026/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 27. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 0, 0], right=[2, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3], [2, 3], [1, 3], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [2, 3], [1, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0027/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 28. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[0, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3, 2], [0, 3], [3], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [0, 3], [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0028/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 29. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([0, 3], [2], [2, 3], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([0, 3], [2], [2, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0029/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 30. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3, 2], [1, 3], [2], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [1, 3], [2], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0030/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 31. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 0, 0], right=[1, 2, 0, 0]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([2], [1, 2], [3, 2], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2], [1, 2], [3, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0031/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 32. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 1, 1], right=[1, 3, 1, 1]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3, 0], [1, 3], [3], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 0], [1, 3], [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0032/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 33. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 1, 1], right=[0, 1, 1, 1]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([1, 3], [0], [0, 3], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [0], [0, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0033/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 34. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[0, 2, 1, 1]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([2, 3], [0, 2], [3], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [0, 2], [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0034/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 35. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 1, 1], right=[0, 3, 1, 1]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3], [0, 3], [2, 3], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [0, 3], [2, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0035/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 36. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[1, 2, 1, 1]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([2, 3], [1, 2], [2], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [1, 2], [2], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0036/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 37. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 1, 1], right=[3, 1, 1, 1]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([1, 2], [3], [3, 2], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [3], [3, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0037/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 38. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 2, 2], right=[1, 0, 2, 2]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([0, 3], [1, 0], [3], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([0, 3], [1, 0], [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0038/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 39. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 2, 2], right=[1, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3], [1, 3], [0, 3], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3], [1, 3], [0, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0039/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 40. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[2, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3, 1], [2, 3], [3], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [2, 3], [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0040/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 41. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 2, 2], right=[1, 2, 2, 2]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([2, 3], [1], [1, 3], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 3], [1], [1, 3], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0041/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 42. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[0, 3, 2, 2]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3, 1], [0, 3], [1], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [0, 3], [1], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0042/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 43. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 2, 2], right=[0, 1, 2, 2]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([1], [0, 1], [3, 1], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1], [0, 1], [3, 1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0043/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 44. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 3, 3], right=[3, 2, 3, 3]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([2, 0], [3, 2], [2], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2, 0], [3, 2], [2], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0044/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 45. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 3, 3], right=[0, 3, 3, 3]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3, 2], [0], [0, 2], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 2], [0], [0, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0045/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 46. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[0, 1, 3, 3]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([1, 2], [0, 1], [2], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [0, 1], [2], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0046/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 47. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 3, 3], right=[0, 2, 3, 3]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([2], [0, 2], [1, 2], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([2], [0, 2], [1, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0047/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 48. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[3, 1, 3, 3]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([1, 2], [3, 1], [1], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 2], [3, 1], [1], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0048/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 49. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 3, 3], right=[2, 3, 3, 3]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([3, 1], [2], [2, 1], 1)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([3, 1], [2], [2, 1], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0049/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 50. Add Polynomial Coefficients with the Parent Ring Table

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0, 0], right=[2, 1, 0, 0, 0]. Give (normalized_left,normalized_right,sum,degree).

View answer and explanation([1, 3], [2, 1], [3], 0)

Normalize each input, pad shorter support with the actual ring zero, add aligned coefficients using the ring addition table and normalize the result. Do not use ordinary arithmetic on carrier labels. The exact result, retaining the requested order, is ([1, 3], [2, 1], [3], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0050/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-add · candidate





## 51. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0], right=[2, 1, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0051/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 52. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 0, 0], right=[2, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3], [2, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3], [2, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0052/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 53. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[0, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3, 2], [0, 3], [0, 2, 3], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 2], [0, 3], [0, 2, 3], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0053/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 54. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 0, 0], right=[2, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([0, 3], [2], [0, 3], 1, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([0, 3], [2], [0, 3], 1, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0054/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 55. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 0, 0], right=[1, 3, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3, 2], [1, 3], [1, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 2], [1, 3], [1, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0055/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 56. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 0, 0], right=[1, 2, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([2], [1, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2], [1, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0056/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 57. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 0, 1, 1], right=[1, 3, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3, 0], [1, 3], [1, 0, 3], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 0], [1, 3], [1, 0, 3], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0057/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 58. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 1, 1], right=[0, 1, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([1, 3], [0], [1, 3], 1, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 3], [0], [1, 3], 1, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0058/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 59. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[0, 2, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([2, 3], [0, 2], [0, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 3], [0, 2], [0, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0059/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 60. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 1, 1], right=[0, 3, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3], [0, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3], [0, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0060/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 61. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 1, 1], right=[1, 2, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([2, 3], [1, 2], [1, 3, 2], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 3], [1, 2], [1, 3, 2], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0061/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 62. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 1, 1], right=[3, 1, 1, 1]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([1, 2], [3], [1, 2], 1, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 2], [3], [1, 2], 1, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0062/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 63. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[0, 3, 2, 2], right=[1, 0, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([0, 3], [1, 0], [1, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([0, 3], [1, 0], [1, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0063/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 64. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 2, 2], right=[1, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3], [1, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3], [1, 3], [3], 0, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0064/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 65. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[2, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3, 1], [2, 3], [2, 1, 3], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 1], [2, 3], [2, 1, 3], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0065/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 66. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 2, 2], right=[1, 2, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([2, 3], [1], [2, 3], 1, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 3], [1], [2, 3], 1, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0066/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 67. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 2, 2], right=[0, 3, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3, 1], [0, 3], [0, 1, 1], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 1], [0, 3], [0, 1, 1], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0067/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 68. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 2, 2], right=[0, 1, 2, 2]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([1], [0, 1], [1], 0, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1], [0, 1], [1], 0, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0068/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 69. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 0, 3, 3], right=[3, 2, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([2, 0], [3, 2], [3, 0, 2], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2, 0], [3, 2], [3, 0, 2], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0069/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 70. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 2, 3, 3], right=[0, 3, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3, 2], [0], [3, 2], 1, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 2], [0], [3, 2], 1, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0070/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 71. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[0, 1, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([1, 2], [0, 1], [0, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 2], [0, 1], [0, 2, 2], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0071/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 72. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[2, 3, 3, 3], right=[0, 2, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([2], [0, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([2], [0, 2], [2], 0, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0072/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 73. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 2, 3, 3], right=[3, 1, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([1, 2], [3, 1], [3, 2, 1], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 2], [3, 1], [3, 2, 1], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0073/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 74. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[3, 1, 3, 3], right=[2, 3, 3, 3]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([3, 1], [2], [3, 1], 1, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0074/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 75. Finite Convolution May Lose Degree over a Ring with Zero Divisors

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. left=[1, 3, 0, 0, 0], right=[2, 1, 0, 0, 0]. Give (normalized_left,normalized_right,product,degree,degree_additivity_not_claimed_with_zero_divisors).

View answer and explanation([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors)

Perform finite convolution using the parent ring multiplication and addition tables, then normalize trailing zeros. Nonzero polynomials can have zero product here; an unconditional additive degree law would be invalid. The exact result, retaining the requested order, is ([1, 3], [2, 1], [2, 3, 3], 2, degree_additivity_not_claimed_with_zero_divisors).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0075/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-multiply · candidate





## 76. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=1. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(1, [0, 1], 2, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (1, [0, 1], 2, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0076/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 77. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=2. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(2, [0, 0, 1], 3, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (2, [0, 0, 1], 3, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0077/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 78. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=3. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(3, [0, 0, 0, 2], 4, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (3, [0, 0, 0, 2], 4, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0078/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 79. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]] and multiplication=[[0, 0, 0, 0], [0, 2, 1, 3], [0, 1, 2, 3], [0, 3, 3, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=4. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(4, [0, 0, 0, 0, 2], 5, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (4, [0, 0, 0, 0, 2], 5, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0079/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 80. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 0, 3, 2], [2, 3, 1, 0], [3, 2, 0, 1]] and multiplication=[[0, 0, 0, 0], [0, 0, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=5. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(5, [0, 0, 0, 0, 0, 3], 6, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (5, [0, 0, 0, 0, 0, 3], 6, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0080/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 81. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 3, 2, 1], [0, 2, 0, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=6. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(6, [0, 0, 0, 0, 0, 0, 3], 7, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (6, [0, 0, 0, 0, 0, 0, 3], 7, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0081/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 82. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 1, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=7. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(7, [1, 1, 1, 1, 1, 1, 1, 0], 8, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (7, [1, 1, 1, 1, 1, 1, 1, 0], 8, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0082/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 83. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[0, 1, 2, 3], [1, 1, 1, 1], [2, 1, 0, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=8. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (8, [1, 1, 1, 1, 1, 1, 1, 1, 0], 9, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0083/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 84. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 2, 3], [0, 1, 3, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=9. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (9, [1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 10, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0084/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 85. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]] and multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=10. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2], 11, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0085/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 86. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 0, 3, 2], [0, 1, 2, 3], [3, 2, 0, 1], [2, 3, 1, 0]] and multiplication=[[1, 1, 0, 0], [1, 1, 1, 1], [0, 1, 3, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=11. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 12, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0086/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 87. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 0, 3, 1], [0, 1, 2, 3], [3, 2, 1, 0], [1, 3, 0, 2]] and multiplication=[[3, 1, 2, 0], [1, 1, 1, 1], [2, 1, 1, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=12. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3], 13, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0087/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 88. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[0, 1, 2, 3], [1, 2, 2, 1], [2, 2, 2, 2], [3, 1, 2, 0]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=13. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (13, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 14, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0088/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 89. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=14. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (14, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0], 15, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0089/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 90. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 1, 2, 3], [2, 2, 2, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=15. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (15, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 16, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0090/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 91. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 0, 1], [2, 3, 1, 0], [0, 1, 2, 3], [1, 0, 3, 2]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 2, 2], [3, 3, 2, 2]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=16. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (16, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], 17, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0091/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 92. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0]] and multiplication=[[2, 0, 2, 0], [0, 3, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=17. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (17, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 18, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0092/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 93. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 3, 0, 2], [3, 2, 1, 0], [0, 1, 2, 3], [2, 0, 3, 1]] and multiplication=[[3, 1, 2, 0], [1, 2, 2, 1], [2, 2, 2, 2], [0, 1, 2, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=18. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (18, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], 19, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0093/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 94. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 3, 1, 3], [2, 1, 0, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=19. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (19, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 20, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0094/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 95. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[0, 1, 2, 3], [1, 0, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=20. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (20, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0], 21, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0095/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 96. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 1, 2, 3], [0, 2, 1, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=21. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (21, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 22, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0096/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 97. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[2, 3, 1, 0], [3, 2, 0, 1], [1, 0, 3, 2], [0, 1, 2, 3]] and multiplication=[[1, 0, 2, 3], [0, 1, 2, 3], [2, 2, 3, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=22. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (22, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1], 23, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0097/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 98. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[3, 2, 1, 0], [2, 0, 3, 1], [1, 3, 0, 2], [0, 1, 2, 3]] and multiplication=[[3, 0, 0, 3], [0, 2, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=23. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (23, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 24, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0098/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 99. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]] and multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=24. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (24, [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2], 25, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0099/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate





## 100. Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial

Use the same qualified complete finite commutative unital ring on Fin(4)={0,…,3}, with addition=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]] and multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 0, 2], [0, 3, 2, 1]] in carrier-label order. All ring axioms and distinct two-sided identities already hold; the zero and one labels need not be 0 and 1. No nonzero-zero-divisor or field premise is silently added. Polynomial coefficients are in increasing exponent order and have finite support. Remove trailing copies of the actual ring zero, representing the zero polynomial by [zero]. The output field degree is the last index of that normalized array, hence 0 for [zero] in this protocol; it is not a claim about the usual degree convention for the zero polynomial. A finite coefficient ring does not make ordinary R[x] finite, and zero divisors forbid an unconditional degree-of-product equality. Nonnegative exponent=25. Give (degree,monomial,coefficient_count,distinct?,R[x]_carrier_infinite_when_zero_ne_one).

View answer and explanation(25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, Yes, R[x]_carrier_infinite_when_zero_ne_one)

Form the coefficient array with the actual ring one at the stated exponent and zero elsewhere. Distinct exponents give distinct monomials because zero differs from one. Each polynomial has finite support even though the collection over all degrees is infinite; this one displayed witness is not itself a finite enumeration of that collection. The exact result, retaining the requested order, is (25, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 26, Yes, R[x]_carrier_infinite_when_zero_ne_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0100/)

Polynomial ring over a finite commutative ring · qf-mod29-poly-infinite-carrier-witness · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations/validation-report.json)