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A finite Galois extension is Galois over every intermediate field

Statement

If $K / F$ is finite Galois and $F \subseteq E \subseteq K$ , then $K / E$ is finite Galois.

Facts & Assumptions

Given: A finite normal and separable extension $K / F$ (Finite Galois extensions and $Gal (K / F)$ ) and an intermediate field $E$ ; separability means every element has a separable minimal polynomial (Separable algebraic elements and separable extensions), and a finite $F$ -basis of $K$ also spans $K$ over $E$ .

[L1]

If $K / F$ is a normal algebraic extension and $F \subseteq E \subseteq K$ , then $K / E$ is a normal algebraic extension (If $K / F$ is normal and $F \subseteq E \subseteq K$ , then $K / E$ is normal).

Proof

technique · direct

1.1

Normality of $K / F$ descends through the intermediate field, so $K / E$ is normal.

1.2

For $α \in K$ , its minimal polynomial over $E$ divides its separable minimal polynomial over $F$ , since the latter lies in $E [x]$ and vanishes at $α$ . A divisor of a separable polynomial is separable, so $K / E$ is separable. This includes $α = 0$ .

givenalgebra

2.1

The extension $K / E$ is finite because a finite $F$ -basis spans it over $E$ ; together with steps 1.1 and 1.2 this makes $K / E$ finite Galois. Both endpoints are included: $E = F$ recovers the hypothesis and $E = K$ gives the degree-one extension.

step 1.1step 1.2given∎

Remarks

The conclusion concerns $K / E$ . The extension $E / F$ need not be normal; the normal-subgroup criterion identifies exactly when it is.

Depends on

Used by

The fundamental theorem of finite Galois theory Theorem

Dependency tree · two levels

9 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, Corollary 3.13 (standard reference, not scraped)