{
  "artifactId": "artifact.library.mathlib.eisenstein-criterion.v001",
  "candidateOnly": false,
  "category": "Algebra and number theory",
  "claimBoundary": "This target indexes Mathlib's prime-ideal Eisenstein criterion for a polynomial over a commutative integral domain. The conclusion requires a prime ideal P, positive degree, primitivity, leading coefficient outside P, every coefficient strictly below the degree inside P, and constant coefficient outside P². It concludes irreducibility in R[X]; it is not an unrestricted prime-element or algorithmic criterion.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem Polynomial.irreducible_of_eisenstein_criterion {R : Type*} [CommRing R] [IsDomain R] {f : R[X]} {P : Ideal R} (hP : P.IsPrime) (hfl : f.leadingCoeff ∉ P) (hfP : ∀ n : ℕ, ↑n < degree f → f.coeff n ∈ P) (hfd0 : 0 < degree f) (h0 : f.coeff 0 ∉ P ^ 2) (hu : f.IsPrimitive) : Irreducible f",
  "family": "Polynomial irreducibility",
  "id": "library.mathlib.eisenstein-criterion.v001",
  "links": {
    "collection": "/collections/landmark-theorems-in-mathlib/",
    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/eisenstein-criterion/",
    "entryData": "/data/library-theorems/eisenstein-criterion.json",
    "evidence": "/proofs/artifact.library.mathlib.eisenstein-criterion.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.eisenstein-criterion.v001.evidence.json",
    "source": "/sources/upstream/eisenstein-criterion/"
  },
  "landmark": {
    "completeFormalType": "theorem Polynomial.irreducible_of_eisenstein_criterion {R : Type*} [CommRing R] [IsDomain R] {f : R[X]} {P : Ideal R} (hP : P.IsPrime) (hfl : f.leadingCoeff ∉ P) (hfP : ∀ n : ℕ, ↑n < degree f → f.coeff n ∈ P) (hfd0 : 0 < degree f) (h0 : f.coeff 0 ∉ P ^ 2) (hu : f.IsPrimitive) : Irreducible f",
    "concepts": [
      "polynomial irreducibility",
      "prime ideals",
      "primitive polynomials",
      "quotient rings",
      "fraction fields",
      "Eisenstein criterion"
    ],
    "plainLanguage": "Choose a prime ideal P in an integral domain. If every coefficient below the leading term lies in P, the leading coefficient stays outside P, the constant coefficient stays outside P², and the polynomial is primitive with positive degree, then the polynomial cannot factor into two nonunits.",
    "poster": {
      "alt": "An ivory and dark-green theorem poster lists the exact prime-ideal, coefficient, positive-degree, and primitive hypotheses, shows the quotient-to-fraction-field route, and concludes that f is irreducible.",
      "byteSize": 2526664,
      "caption": "A primitive positive-degree polynomial satisfying the stated P and P² coefficient conditions is irreducible.",
      "description": "The poster centers the exact hypotheses above a coefficient row with the leading coefficient outside P and constant coefficient outside P². A sparse route below passes through R/P and FractionRing(R/P), where the lower coefficients vanish and generalized Eisenstein is applied at q = X.",
      "derivatives": [
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          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/eisenstein-criterion/theorem-poster-v1-640.webp",
          "sha256": "sha256:cc48860784c72cac16e2dba92c07c22ac1a2fa7e42b9121569013b08491db7ba",
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        {
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          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/eisenstein-criterion/theorem-poster-v1-1024.webp",
          "sha256": "sha256:b879b73ec2592d3b4c1a00654cfd48c3f4a6e46ea305629ab248fd4d9e9667b6",
          "width": 1024
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      ],
      "generatedAt": "2026-07-25T20:30:21Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/eisenstein-criterion/theorem-poster-v1.png",
      "sha256": "sha256:87f253f89c88b7caabf45a27c2329b132632ff8723ca1fc730c7805152576d99",
      "title": "Eisenstein Criterion at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "ALGEBRA · POLYNOMIALS\nEISENSTEIN CRITERION\nR an integral domain · P a prime ideal · f in R[X]\nleading coefficient outside P\nevery lower coefficient inside P\npositive degree · f primitive\nconstant coefficient outside P²\nf is irreducible\nTHE CHECKED ROUTE\nR → R/P → FractionRing(R/P)\nlower coefficients vanish · a pure monomial remains\ngeneralized Eisenstein at q = X\nEXACT SCOPE\nPrime-ideal and primitive formulation; no principal ideal, UFD, field, monic, or algorithm claim."
    },
    "visual": {
      "alt": "A dark-green engraved schematic places all lower polynomial coefficients inside a prime-ideal enclosure, the leading coefficient outside it, and the constant coefficient inside P but outside a nested P-squared enclosure before passing through a quotient and fraction-field lens to a forced trivial factorization.",
      "byteSize": 2857095,
      "caption": "The prime-ideal coefficient pattern survives quotient and fraction-field passage as a pure monomial, while the constant term's exclusion from P² blocks a nontrivial factorization.",
      "description": "A coefficient spine enters a large ideal enclosure with its leading bead left outside and its constant bead separated from a smaller nested enclosure. A quotient aperture and fraction-field lens erase the lower coefficients, and a tentative two-branch factorization returns to the P² obstruction before closing one branch as a unit.",
      "derivatives": [
        {
          "byteSize": 45358,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/eisenstein-criterion/theorem-schematic-v1-640.webp",
          "sha256": "sha256:485574c5074b6936cf3c0821f1cba05f336c4daede1d1a8942aec9d7099c63d2",
          "width": 640
        },
        {
          "byteSize": 173382,
          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/eisenstein-criterion/theorem-schematic-v1-1200.webp",
          "sha256": "sha256:b0a4b7df8ced7199a02b43a32141d740062ad1cc958037d7323770d211c0e41e",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T20:30:21Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/eisenstein-criterion/theorem-schematic-v1.png",
      "sha256": "sha256:9576be5e4fbba355b7f03818acf37b2c178f2991a7994206dca2c1634c275c60",
      "title": "Eisenstein criterion schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Eisenstein's criterion is a fundamental and widely used test for polynomial irreducibility. Mathlib states it in a general prime-ideal form and derives it from a generalized quotient-and-fraction-field criterion, making the exact role of primitivity and the P² obstruction explicit."
  },
  "nonClaims": [
    "Proof Atlas did not originate Eisenstein's criterion or Mathlib's declaration.",
    "The selected declaration does not assume or conclude that P is principal or maximal, R is a UFD or field, f is monic, or its leading coefficient is a unit.",
    "The theorem does not provide a factorization or irreducibility algorithm, and it does not remove the primitivity or positive-degree hypotheses.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.eisenstein-criterion.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "A primitive positive-degree polynomial over an integral domain is irreducible when a prime ideal contains every lower coefficient but not the leading coefficient, while its square does not contain the constant coefficient.",
  "targetId": "target.library.mathlib.eisenstein-criterion.v001",
  "title": "Eisenstein Criterion",
  "upstreamOrigin": {
    "declarationName": "Polynomial.irreducible_of_eisenstein_criterion",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/RingTheory/Polynomial/Eisenstein/Criterion.lean#L175",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
    "sourceArtifactHash": "sha256:0faf5ccbec545ac707711ff9cec63349de336c49d176f64f67b93f8c03feef29",
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    "sourceFile": "Mathlib/RingTheory/Polynomial/Eisenstein/Criterion.lean",
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    "verificationKind": "git_worktree"
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  "publicPresentationReview": {
    "status": "reviewed"
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}