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.

$\blacksquare$