{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261031-r06-finite-field-0070", "pilot_number": "R0066-33620", "number": 33620, "family_id": "qf-mod31-fielddeep-affine-solve", "title": { "zh-Hant": "第 33620 題：有限域深度", "en": "Question 33620: Solve a Nonzero-Coefficient Affine Equation in the Prime Field" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261031-r06-finite-field-0070", "batch_id": "batch-20261014-advance-20261031-r06-finite-field-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-field" ], "title_zh": "有限域深度", "stem_zh": "完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":71,\"a\":20,\"b\":20,\"c\":39}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。", "answer": { "kind": "expression", "value": "(20, 20, 39, 32, 40, 39, 是)", "explanation_zh": "Wave4收束核對=(20, 20, 39, 32, 40, 39, 是)；依完整明示資料重算；只對非零元素取逆且永不除以零；有限域階為prime power但本題族不宣稱每個有限環為域。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限域深度" ] }, "variant": { "family_id": "qf-mod31-fielddeep-affine-solve", "seed": 202610316070, "parameters": { "p": 71, "a": 20, "b": 20, "c": 39, "domain": "Exact_prime_field_Zp", "operation": "fielddeep-affine-solve", "semantic_template": "qf-mod31-fielddeep-affine-solve/{\"a\":20,\"b\":20,\"c\":39,\"p\":71}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般不可數性／色數最優性／更強機率收斂／商域、optional stopping、Lean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-20261031-r06-finite-field-0070", "batch_id": "batch-20261014-advance-20261031-r06-finite-field-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "0ce6690ebb8e3842a85d439f60d42e1704f2be6fafe5a9fdaf8a89831873e784", "family_id": "qf-mod31-fielddeep-affine-solve", "seed": 202610316070, "semantic_template": "qf-mod31-fielddeep-affine-solve/{\"a\":20,\"b\":20,\"c\":39,\"p\":71}" }, "fields": { "title": "Solve a Nonzero-Coefficient Affine Equation in the Prime Field", "stem": "Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=20, b=20, c=39; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).", "answer": { "kind": "expression", "value": "(20, 20, 39, 32, 40, 39, Yes)", "explanation": "Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (20, 20, 39, 32, 40, 39, Yes)." }, "skills": [ "Solve a Nonzero-Coefficient Affine Equation in the Prime Field" ], "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": "完整prime field Z_p運算。操作=fielddeep-affine-solve；資料={\"p\":71,\"a\":20,\"b\":20,\"c\":39}。請依 [a,b,c,inverse,solution,substitution,unique] 欄序回答。", "en": "Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=20, b=20, c=39; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod31-fielddeep-affine-solve, [a, b, c, domain, operation, p]]", "after": "derived = [fielddeep-affine-solve, Exact_prime_field_Zp]", "relation": "implies", "rule_id": "R66_EXACT_source_bank_keys_local_operation_domain", "conditions": [], "explanation": { "zh-Hant": "原bank的operation是LOCAL hyphen，keys與domain各自精確匹配。欄位順序及數字/Boolean/字串角色依來源，所有有理數文字整數省略/1；不改原題與英文、MKO、AST或狀態。", "en": "THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [71, [2, 3, 4, 5, 6, 7, 8]]", "after": "derived = [[1, 2, 3, 1, 5, 1, 7], 是]", "relation": "implies", "rule_id": "Field_PRIME_complete_trial_divisor_qualification", "conditions": [], "explanation": { "zh-Hant": "完整試除2到floor(sqrt(p))無因數；若p合成必有其中因數。2也合法。只限定本題Z_p，不把一般有限域都說成prime order或把有限環都說成域。", "en": "Trial EVERY divisor2 through floor(sqrt(p));a composite would have a factor there.The empty test for p=2 is valid.This qualifies THIS prime field Z_p,not every finite field’s order or every finite ring as a field." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。", "en": "Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Field ORIGINAL a times proposed inverse := mul(20, 32)", "after": "Field ORIGINAL a times proposed inverse = 640", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本步原操作數與次序精確運算；來源有理數的pre-reduction界另行保留，不用浮點或偷偷跨bank放寬。", "en": "Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Field ORIGINAL a times proposed inverse，不以小數近似替代。", "en": "Recompute Field ORIGINAL a times proposed inverse with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "20", "32" ], "output": "640", "label": "Field ORIGINAL a times proposed inverse" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [inverse substitution, 640, 71]", "after": "derived = [9, 1]", "relation": "implies", "rule_id": "Finite_MOD_euclidean_quotient_remainder", "conditions": [], "explanation": { "zh-Hant": "以整數Euclidean division核value=n*quotient+remainder與0≤remainder