{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261031-r06-finite-field-0033", "pilot_number": "R0066-33683", "number": 33683, "family_id": "qf-mod31-fielddeep-power-cycle", "title": { "zh-Hant": "第 33683 題：有限域深度", "en": "Question 33683: The Nonzero Multiplicative Group Gives a p−1 Power Identity" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261031-r06-finite-field-0033", "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-power-cycle；資料={\"p\":19,\"element\":8}。請依 [element,powers,last,is_one,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "(8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, 是, nonzero_multiplicative_group_cycle)", "explanation_zh": "Wave4收束核對=(8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, 是, nonzero_multiplicative_group_cycle)；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": 1, "skills": [ "有限域深度" ] }, "variant": { "family_id": "qf-mod31-fielddeep-power-cycle", "seed": 202610316033, "parameters": { "p": 19, "element": 8, "domain": "Exact_prime_field_Zp", "operation": "fielddeep-power-cycle", "semantic_template": "qf-mod31-fielddeep-power-cycle/{\"element\":8,\"p\":19}" } }, "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-0033", "batch_id": "batch-20261014-advance-20261031-r06-finite-field-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "5b67e5310e548d261a9cf6f58ce0bac8e84d4c7d10db38a9848b8e0d2c1da345", "family_id": "qf-mod31-fielddeep-power-cycle", "seed": 202610316033, "semantic_template": "qf-mod31-fielddeep-power-cycle/{\"element\":8,\"p\":19}" }, "fields": { "title": "The Nonzero Multiplicative Group Gives a p−1 Power Identity", "stem": "Use the complete prime field Z_19, with prime p=19, 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 element=8. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.", "answer": { "kind": "expression", "value": "(8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, Yes, nonzero_multiplicative_group_cycle)", "explanation": "Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, Yes, nonzero_multiplicative_group_cycle)." }, "skills": [ "The Nonzero Multiplicative Group Gives a p−1 Power Identity" ], "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-power-cycle；資料={\"p\":19,\"element\":8}。請依 [element,powers,last,is_one,certificate] 欄序回答。", "en": "Use the complete prime field Z_19, with prime p=19, 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 element=8. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-mod31-fielddeep-power-cycle, [domain, element, operation, p]]", "after": "derived = [fielddeep-power-cycle, 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 = [19, [2, 3, 4]]", "after": "derived = [[1, 1, 3], 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 previous power times element k=1 := mul(1, 8)", "after": "Field ORIGINAL previous power times element k=1 = 8", "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 previous power times element k=1，不以小數近似替代。", "en": "Recompute Field ORIGINAL previous power times element k=1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "8" ], "output": "8", "label": "Field ORIGINAL previous power times element k=1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [power exponent1, 8, 19]", "after": "derived = [0, 8]", "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