TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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Finite separable extensions have finite minimal Galois closures

Statement

Let $K / F$ be a finite separable extension inside an algebraic closure $Ω$ of $F$ . Its Galois closure $L = N_{Ω} (K / F)$ exists, is finite Galois over $F$ , and is contained in every subfield of $Ω$ that contains $K$ and is Galois over $F$ .

Facts & Assumptions

Given: A finite separable extension $K / F \subseteq Ω$ , the Galois-closure definition of The Galois closure of a finite separable extension, the fact that an algebraic extension generated by separable elements is separable (An algebraic extension generated by separable elements is separable), and the equivalence between finite Galois extensions and separable splitting fields (Equivalent characterizations of a finite Galois extension).

[L1]

The normal closure in $Ω$ of a finite extension is finite over $F$ and is the splitting field of the product of the minimal polynomials of a finite generating family (The normal closure of a finite extension exists and is finite).

Proof

technique · direct

1.1

Choose a finite generating family $K = F (α_{1}, \dots, α_{r})$ . By [L1], the normal closure $L$ is finite over $F$ and is the splitting field of the product of the minimal polynomials of the generators. If $r = 0$ , then $K = F$ and $L = F$ . Repeated minimal polynomials may be removed from the product.

2.1

Each generator is separable over $F$ , so every root of its minimal polynomial is separable over $F$ . The field $L$ is generated by those roots, hence is separable; it is normal by construction, so it is finite Galois.

step 1.1given

3.1

If $M$ is a subfield of $Ω$ containing $K$ and Galois over $F$ , then $M / F$ is normal. The normal closure is the intersection of all normal subextensions containing $K$ , so $L \subseteq M$ . Thus $L$ is the required minimal Galois overfield.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by The Galois closure of a finite separable extension.

Dependency tree · two levels

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Sources

J. S. Milne, Fields and Galois Theory, v5.10, Corollary 3.12 and Remark 3.18 (standard reference, not scraped)
K. Conrad, The Galois Correspondence, Theorem 4.8 (standard reference, not scraped)