{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-field",
  "semantic_family_id": "sf-finite-field",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限域"
  },
  "summary": {
    "zh-Hant": "有限交換含一環中每個非零元素皆有乘法逆元時構成有限域。"
  },
  "statement": {
    "zh-Hant": "以完整有限環雙表核1不等於0、乘法交換及每個非零元素的雙側逆元。除法只對非零除數定義。有限整環成域是利用有限乘法映射單射推出滿射的定理步驟；域階為質數冪，不必是質數。"
  },
  "formula": {
    "tex": "F(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>F</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "F",
        "glyph": "F"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "227f53786ce57d2fe3de0c4c8b3ae24c4e2af89efeadd6d69dd32e1e61a87fca"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限交換含一環表及零、一明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "逆元逐全部非零元素核對；永不除以零。"
    }
  ],
  "symbols": [
    {
      "token": "F",
      "role": "finite_field",
      "role_zh": "有限域",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "finite_integral_domain",
      "role_zh": "父有限整環",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-integral-domain",
      "title_zh": "有限整環",
      "reason_zh": "有限性定理提供非零逆元。"
    },
    {
      "id": "mko-finite-unital-ring",
      "title_zh": "有限含一環",
      "reason_zh": "保留0、1與完整雙表。"
    },
    {
      "id": "mko-finite-abelian-group",
      "title_zh": "有限阿貝爾群",
      "reason_zh": "加法群結構完整。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整搜尋每個非零元逆元。",
      "有限整環到域保留單射／滿射定理步驟。",
      "有限域階可為p的冪，不限質數。"
    ],
    "common_misconception_zh": "把任意有限交換環當域、允許除以零，或宣稱有限域階一定是質數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只分類指定完整有限表；不外推任意交換環或省略非零逆元責任。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入有限域上的多項式與擴張。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-field.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-10-30 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}