# Question 14735: Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial [Return to catalogue page 74](https://ocme.evemisslab.com/en/questions/page/74/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations/#q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0085) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0028-14735 · q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0085 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-polynomial-ring-over-finite-commutative-ring](https://ocme.evemisslab.com/en/mko/mko-polynomial-ring-over-finite-commutative-ring/) ## Step-by-step derivation S001 · impliesBefore same carrier Fin(4);addition=[[3, 0, 1, 2], [0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1]];multiplication=[[2, 1, 0, 3], [1, 1, 1, 1], [0, 1, 2, 3], [3, 1, 3, 1]] After actual additive zero=1;actual multiplicative one=2;additive inverses=[2, 1, 0, 3] Conditions: none Bind coefficients and domain reasoning to the same complete parent tables.Identities are actual elements,not presumed labels0 or1. Step identifiers and dependenciesRule: commutative_parent_actual_tables_and_identities · Uses: G001 S002 · impliesBefore all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null] After additive commute=Yes;multiplicative commute=Yes;zero differs from one=Yes Conditions: none Qualify the whole additive abelian group,multiplicative associativity/commutativity,both distributive laws and distinct identities.Domain and polynomial obligations follow this premise,not sampled successes. Step identifiers and dependenciesRule: complete_commutative_unital_parent_qualification · Uses: G001, S001 S003 · calculationBefore monomial coefficient count := sum(10, 1) After monomial coefficient count = 11 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore arbitrary nonnegative exponent;displayed degree=10;actual zero=1;one=2 After monomial coefficients=[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2];coefficient count=11 Conditions: none This polynomial has finite support,with one at the selected exponent and the same parent zero elsewhere.It is not one infinitely long polynomial. Step identifiers and dependenciesRule: polynomial_each_monomial_finite_support · Uses: G001, S003 S005 · impliesBefore zero=1 differs from one=2;compare any two different exponents d