Existing in Mathlib · Algebra

Primitive Element Theorem

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.

field extensions
primitive elements
finite-dimensional extensions
separable extensions
simple extensions
intermediate fields

Exact theorem

Exact Mathlib statement

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⟯ = ⊤

The theorem at a glance

Primitive element theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

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. — A finite-dimensional separable field extension is simple: one element generates the whole extension over the base field.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Three generators x, y, and z in E/F pass through one funnel and emerge as a single gold generator alpha with E equal to F(alpha). — A finite-dimensional separable extension is generated by one element.

Why it matters

A mathematical landmark

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.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

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.

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.

Source and local evidence

Where the theorem comes from

Existing declaration: Field.exists_primitive_element in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.primitive-element-theorem.v001
Source: Open the pinned upstream reference