[[Field theory MOC]]
# Galois extension

An extension $F: K$ is **Galois** iff it is [[Separable extension|separable]] and [[Normal extension|normal]]. #m/def/field/gal

## Finite Galois extension

Let $F:K$ be a [[Finite field extension|finite extension]]. Then the following are equivalent:[^2009]

1. $F:K$ is Galois;
2. $F$ is the [[splitting field]] of a [[separable polynomial]] $f(x) \in K[x]$ over $K$;
3. $F:K$ is [[separable extension|separable]] and [[Normal extension|normal]];
4. $\abs{\Aut(F:K)} = [F:K]$;
5. $K = F^{\Aut(F:K)}$ is the [[Fixed field of an automorphism group|fixed field]] for $\Aut(F:K)$;
6. the [[Fixed field of an automorphism group|Galois correspondence]] for $F:K$ is a [[Surjectivity, injectivity, and bijectivity|bijection]];
7. $F:K$ is [[Separable extension|separable]], and if $L:F$ is an [[Algebraic element|algebraic extension]] and $\sigma \in \Aut(L:K)$, then $\sigma(F) = F$.

  [^2009]: 2009\. [[Sources/@aluffiAlgebraChapter02009|Algebra: Chapter 0]], §VII.6.1, p. 457

## Properties

- Every extension of a [[Galois field]] is Galois cyclic (generated by the [[Frobenius endomorphism|Frobenius automorphism]])

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