DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The normal closure of an algebraic extension inside a fixed algebraic closure

Definition

Let $F \subseteq K \subseteq Ω$ , where $K / F$ is algebraic and $Ω / F$ is a fixed algebraic closure (An algebraic closure of a field). The normal closure of $K / F$ in $Ω$ is

$N_{Ω} (K / F) : = ⋂ {E : K \subseteq E \subseteq Ω and E / F is normal} .$

The family being intersected is nonempty: $Ω / F$ is normal because every minimal polynomial over $F$ splits in the algebraically closed field $Ω$ (A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there). Its intersection is normal by A nonempty intersection of normal subextensions inside a common algebraic extension is normal, so the definition produces the smallest normal intermediate extension containing $K$ .

Depends on

Used by

The Galois closure of a finite separable extension Definition
The normal closure of a finite extension exists and is finite Theorem

Dependency tree · two levels

8 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

The Stacks Project, Section 9.15: Normal extensions (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapter 6 (standard reference, not scraped)