{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0099", "pilot_number": "R0028-14749", "number": 14749, "family_id": "qf-mod29-poly-infinite-carrier-witness", "title": { "zh-Hant": "第 14749 題：有限交換環上的多項式環", "en": "Question 14749: Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0099", "batch_id": "batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-polynomial-ring-over-finite-commutative-ring" ], "title_zh": "有限交換環上的多項式環", "stem_zh": "係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"degree\":24}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(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, 是, R[x]_carrier_infinite_when_zero_ne_one)", "explanation_zh": "Wave4第三層核對=(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, 是, R[x]_carrier_infinite_when_zero_ne_one)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；普通R[x]載體不因係數環有限而有限；有零因子時不宣稱degree乘法加法或域性質。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限交換環上的多項式環" ] }, "variant": { "family_id": "qf-mod29-poly-infinite-carrier-witness", "seed": 202610292099, "parameters": { "n": 4, "addition": [ [ 1, 2, 3, 0 ], [ 2, 3, 0, 1 ], [ 3, 0, 1, 2 ], [ 0, 1, 2, 3 ] ], "multiplication": [ [ 2, 1, 0, 3 ], [ 1, 3, 1, 3 ], [ 0, 1, 2, 3 ], [ 3, 3, 3, 3 ] ], "degree": 24, "domain": "Finite_support_polynomial_over_finite_commutative_ring", "operation": "poly-infinite-carrier-witness", "semantic_template": "qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"degree\":24,\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-0099", "batch_id": "batch-20261014-advance-20261029-r02-polynomial-ring-over-finite-commutative-ring-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "ac932b9e6e2c07039b0934ab93fea907bdf7cbff9ee5c2d68807fbf6c30de108", "family_id": "qf-mod29-poly-infinite-carrier-witness", "seed": 202610292099, "semantic_template": "qf-mod29-poly-infinite-carrier-witness/{\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"degree\":24,\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"n\":4}" }, "fields": { "title": "Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial", "stem": "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).", "answer": { "kind": "expression", "value": "(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)", "explanation": "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)." }, "skills": [ "Unbounded Exponents, Not Infinitely Many Coefficients in One Polynomial" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "係數屬同一完整有限交換環且多項式有限支撐。操作=poly-infinite-carrier-witness；資料={\"n\":4,\"addition\":[[1,2,3,0],[2,3,0,1],[3,0,1,2],[0,1,2,3]],\"multiplication\":[[2,1,0,3],[1,3,1,3],[0,1,2,3],[3,3,3,3]],\"degree\":24}。請依 [degree,monomial,coefficient_count,distinct,boundary] 欄序回答。", "en": "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)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "same carrier Fin(4);addition=[[1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2], [0, 1, 2, 3]];multiplication=[[2, 1, 0, 3], [1, 3, 1, 3], [0, 1, 2, 3], [3, 3, 3, 3]]", "after": "actual additive zero=3;actual multiplicative one=2;additive inverses=[2, 1, 0, 3]", "relation": "implies", "rule_id": "commutative_parent_actual_tables_and_identities", "conditions": [], "explanation": { "zh-Hant": "所有係數與整環論證綁同一完整父表；單位元是實際元素，不預設標籤0或1。", "en": "Bind coefficients and domain reasoning to the same complete parent tables.Identities are actual elements,not presumed labels0 or1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "all additive/multiplicative pairs and triples checked;first failures=[null, null, null, null]", "after": "additive commute=是;multiplicative commute=是;zero differs from one=是", "relation": "implies", "rule_id": "complete_commutative_unital_parent_qualification", "conditions": [], "explanation": { "zh-Hant": "完整核加法Abelian群、乘法結合與交換、左右分配及0≠1；整環與多項式責任在此合格前提之後，不用有限個正樣本冒充全表資格。", "en": "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." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "monomial coefficient count := sum(24, 1)", "after": "monomial coefficient count = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已核中間量精確計算，保留分母、正負號、索引及必要條件。", "en": "Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and necessary guards." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 monomial coefficient count，不以小數近似替代。", "en": "Recompute monomial coefficient count with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "24", "1" ], "output": "25", "label": "monomial coefficient count" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "arbitrary nonnegative exponent;displayed degree=24;actual zero=3;one=2", "after": "monomial coefficients=[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];coefficient count=25", "relation": "implies", "rule_id": "polynomial_each_monomial_finite_support", "conditions": [], "explanation": { "zh-Hant": "這個多項式只有有限支撐，one放在指定指數，其餘是同一父環實際零；不生成無窮長的一個多項式。", "en": "This polynomial has finite support,with one at the selected exponent and the same parent zero elsewhere.It is not one infinitely long polynomial." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "zero=3 differs from one=2;compare any two different exponents d