Automorphism Group Acts Faithfully on Generating Set
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Theorem
Let $E/F$ be a field extension.
Let $\operatorname{Aut}(E/F)$ be its automorphism group.
Let $S\subset E$ be a generating set of the extension.
Let $S$ be stable under the group action of $\operatorname{Aut}(E/F)$.
Then the induced group action on $S$ is faithful.
Proof
Let $\sigma \in \operatorname{Aut}(E/F)$ stabilize $S$.
Then $S$ is contained in the fixed field of $\sigma$.
By definition of generating set, $\sigma$ fixes $E$.
Thus $\operatorname{Aut}(E/F)$ acts faithfully.
$\blacksquare$