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).
G001 — 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).
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.
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.
zero=3 differs from one=2;compare any two different exponents d<e
After
coefficient at e distinguishes x^e from x^d;ordinary R[x] carrier infinite
Conditions: none
For arbitrary d<e,the coefficient at e is one in one monomial and zero in the other,so monomials at distinct exponents differ.A finite coefficient ring does not make ordinary R[x] finite,nor does one displayed witness enumerate it.
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.