{
  "artifactId": "artifact.library.mathlib.primitive-element-theorem.v001",
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  "category": "Algebra",
  "claimBoundary": "This target indexes Mathlib's existence theorem for a primitive element of a finite-dimensional separable field extension E/F: some α : E satisfies F⟮α⟯ = ⊤. Finite means finite-dimensional, not finite cardinality. The result does not assert uniqueness or canonicity of α and does not cover general inseparable extensions.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem Field.exists_primitive_element (F E : Type*) [Field F] [Field E] [Algebra F E] [FiniteDimensional F E] [Algebra.IsSeparable F E] : ∃ α : E, F⟮α⟯ = ⊤",
  "family": "Field theory",
  "id": "library.mathlib.primitive-element-theorem.v001",
  "links": {
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    "entry": "/library-theorems/primitive-element-theorem/",
    "entryData": "/data/library-theorems/primitive-element-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.primitive-element-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.primitive-element-theorem.v001.evidence.json",
    "source": "/sources/upstream/primitive-element-theorem/"
  },
  "landmark": {
    "completeFormalType": "theorem Field.exists_primitive_element (F E : Type*) [Field F] [Field E] [Algebra F E] [FiniteDimensional F E] [Algebra.IsSeparable F E] : ∃ α : E, F⟮α⟯ = ⊤",
    "concepts": [
      "field extensions",
      "primitive elements",
      "finite-dimensional extensions",
      "separable extensions",
      "simple extensions",
      "intermediate fields"
    ],
    "plainLanguage": "If E is a finite-dimensional separable extension of a field F, then one can choose an element α of E such that adjoining α to F produces all of E. The generator exists, but the theorem does not say that it is unique or canonical.",
    "poster": {
      "alt": "An ivory and dark-green field-theory poster explains that a finite-dimensional separable extension has one generator and shows the finite and infinite base-field proof routes converging.",
      "byteSize": 2903663,
      "caption": "A finite-dimensional separable field extension is simple: one element generates the whole extension over the base field.",
      "description": "The poster states the finite-dimensional separable hypotheses, contrasts the cyclic-generator route for finite base fields with simple-adjoin compression over infinite base fields, brings both routes to one gold generator, and closes with the theorem's exact non-uniqueness and non-cardinality boundary.",
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      "generatedAt": "2026-07-25T16:20:34-04:00",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/primitive-element-theorem/theorem-poster-v1.png",
      "sha256": "sha256:5d1b8af5035f76d23f3fb9546b993db8731efd12c8438993662114d7162dae6f",
      "title": "Primitive element theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "ALGEBRA · FIELD THEORY\nPRIMITIVE ELEMENT THEOREM\nONE ELEMENT GENERATES THE WHOLE EXTENSION\nHYPOTHESES\nE is finite-dimensional and separable over F\nTWO ROUTES TO ONE GENERATOR\nFinite base field · choose a cyclic generator\nInfinite base field · compress two adjunctions into one\nRepeat the compression until one element remains\nCONCLUSION\nSome element of E generates all of E over F\nEXACT SCOPE\nFinite means finite-dimensional, not finite cardinality\nNo unique or canonical generator is asserted"
    },
    "visual": {
      "alt": "A text-free engraved field diagram shows a finite cyclic rosette and repeated pairwise compression converging on one gold generator whose nested contours fill the extension.",
      "byteSize": 3231412,
      "caption": "The finite- and infinite-base-field proof routes both reach a single element that generates the full finite-dimensional separable extension.",
      "description": "A dark-green base-field chamber supports two distinct proof corridors. The finite-base-field corridor passes through a cyclic rosette, while the infinite-base-field corridor repeatedly compresses paired generators and avoids finitely many obstructions. Both arrive at one luminous generator surrounded by nested contours that fill the upper extension chamber.",
      "derivatives": [
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          "publicPath": "/assets/library-theorems/primitive-element-theorem/theorem-schematic-v1-640.webp",
          "sha256": "sha256:080dd84e0ae5a9bcb0d6b3aa6e8ca1ae08df51c4b2fc3a58fa45c23682f3df65",
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          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/primitive-element-theorem/theorem-schematic-v1-1200.webp",
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          "width": 1200
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      ],
      "generatedAt": "2026-07-25T16:20:34-04:00",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/primitive-element-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:eb10c32ff365f35609298eae715f80097331e281d543ef706bee9b4266818c74",
      "title": "Two routes to one primitive element",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "The primitive element theorem compresses a finite-dimensional separable extension from many possible generators to one. It is a foundational bridge between abstract field extensions and the concrete study of a single minimal polynomial."
  },
  "nonClaims": [
    "Proof Atlas did not originate the primitive element theorem or Mathlib's proof.",
    "The theorem does not choose a unique, canonical, or computationally preferred generator.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.primitive-element-theorem.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": "Every finite-dimensional separable field extension is generated over its base field by a single element.",
  "targetId": "target.library.mathlib.primitive-element-theorem.v001",
  "title": "Primitive Element Theorem",
  "upstreamOrigin": {
    "declarationName": "Field.exists_primitive_element",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/FieldTheory/PrimitiveElement.lean#L213",
    "packageName": "mathlib",
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    "sourceFile": "Mathlib/FieldTheory/PrimitiveElement.lean",
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    "verificationKind": "git_worktree"
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  "publicPresentationReview": {
    "status": "reviewed"
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}