Finite Field Extension has Finite Galois Group

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Theorem

Let $E / F$ be a finite field extension.


Then its automorphism group is finite.


Proof

Because $E / F$ is finite, it is finitely generated.

Let $\alpha_1, \ldots, \alpha_n \in E$ with $E = \map F {\alpha_1, \ldots, \alpha_n}$.



By Finite Field Extension is Algebraic, $\alpha_1, \ldots, \alpha_n$ are algebraic over $F$.

Let $f_1, \ldots, f_n$ be their minimal polynomials.

Let $f = f_1\dots f_n$.

By Galois Group Acts Faithfully on Generating Set, $\Gal {E / F}$ acts faithfully on the roots of $f$.

Thus $\Gal {E / F}$ is finite.

$\blacksquare$