DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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The Galois closure of a finite separable extension

Definition

Let $K / F$ be a finite separable extension embedded in a fixed algebraic closure $Ω$ of $F$ . Its Galois closure in $Ω$ is the normal closure $N_{Ω} (K / F)$ of The normal closure of an algebraic extension inside a fixed algebraic closure.

Equivalently, it is the least subfield $L$ of $Ω$ containing $K$ for which $L / F$ is finite Galois (Finite Galois extensions and $Gal (K / F)$ ). The existence, finiteness, separability, and leastness asserted by this terminology are established in Finite separable extensions have finite minimal Galois closures ↗.

Depends on

Used by

Finite separable extensions have finite minimal Galois closures Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory, v5.10, Remark 3.18 (standard reference, not scraped)