# Question 14777: Normalize a Finite-Support Coefficient Array [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-0002) 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. R0028-14777 · q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0002 ## Original question and givens 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). - G001 — 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). ## 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=[[0, 1, 2, 3], [1, 3, 0, 2], [2, 0, 3, 1], [3, 2, 1, 0]];multiplication=[[0, 0, 0, 0], [0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 3, 0]] After actual additive zero=0;actual multiplicative one=1;additive inverses=[0, 2, 1, 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 · impliesBefore given polynomial:raw ascending coefficient labels=[3, 0, 0, 0];actual ring zero=0 After normalized finite support=[3];last-index degree field=0 Conditions: none Remove only trailing actual-zero coefficients,not low-degree or internal zeros.Empty/all-zero inputs become[zero].Here degree means last index and returns0 for the zero polynomial,not the usual mathematical zero-polynomial degree convention. Step identifiers and dependenciesRule: polynomial_trailing_actual_zero_protocol · Uses: G001, S002 ## Reverse checks - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([3, 0, 0, 0], 0, [3], 0) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 085563d2dfdaa54a669d9237098b28bb26f1eb72c5de051a152380a831260081 - candidate record SHA-256: d2e099e9024620c23c70dc4c608dd5184dc6e696835663fcc0a442a7e37c9bf6 - semantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae - question-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0002.json)