Finite Field Extension has Finite Galois Group

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 $E=F(\alpha_1,\ldots,\alpha_n)$.

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, $\operatorname{Gal}(E/F)$ acts faithfully on the roots of $f$.

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