{
  "id": "field",
  "domain": "algebra",
  "type": "Concept",
  "title": "Field",
  "slug": "field",
  "url": "/mathematics/algebra/field/",
  "summary": "A commutative ring where every non-zero element has a multiplicative inverse. The rational, real, and complex numbers are fields of characteristic 0; 𝔽ₚ and GF(pⁿ) are finite fields.",
  "formal_definition": "A commutative ring (F, +, ·) where (F\\{0}, ·) is an abelian group. Equivalently: all four arithmetic operations are defined (division by non-zero).",
  "key_results": [
    { "result": "Classification of finite fields", "statement": "For every prime p and positive integer n, there is exactly one field of order pⁿ up to isomorphism. No other finite fields exist." },
    { "result": "Fundamental Theorem of Algebra", "statement": "ℂ is algebraically closed: every non-constant polynomial over ℂ has a root in ℂ." },
    { "result": "Fundamental Theorem of Galois Theory", "statement": "Subgroups of Gal(K/F) correspond bijectively, order-reversingly, to intermediate fields of K/F." }
  ],
  "introduced_by": "Richard Dedekind, term 'Körper' (body/field), 1871",
  "added_to_codex": "2026-05-27",
  "last_verified": "2026-05-27",
  "sources": [
    { "tier": 1, "citation": "Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra. 3rd ed. John Wiley & Sons.", "chapters": "13–14" },
    { "tier": 1, "citation": "Lang, S. (2002). Algebra. Revised 3rd ed. Springer.", "chapters": "V–VI" },
    { "tier": 2, "citation": "Stewart, I. (2015). Galois Theory. 4th ed. CRC Press." },
    { "tier": 2, "citation": "Lidl, R. and Niederreiter, H. (1994). Introduction to Finite Fields and their Applications. Revised ed. Cambridge University Press." }
  ],
  "relationships": [
    { "type": "specialises", "target": "ring", "label": "Ring" },
    { "type": "specialises", "target": "group", "label": "Group" },
    { "type": "key_person", "target": "dedekind-richard", "label": "Richard Dedekind" },
    { "type": "key_person", "target": "galois-evariste", "label": "Évariste Galois" }
  ],
  "entry_status": "live",
  "reading_levels_available": ["curious", "exploring", "deep_dive"],
  "completeness": 1.0,
  "codex_version": "1.1"
}