If $L/K$ is a field extension, then $L$ can be considered as a $K$-vectorspace of dimension $[L:K]$.
If we consider $K$-automorphisms of $L$, they take $\overline\sigma: L\to L$ where $\overline\sigma|_K = \sigma:K\to K$
This means that $\overline\sigma$ must fix all points in $K$ right? I.e. any automorphisms of an infinite field $K$ must be the identity map? $\overline \sigma: a\mapsto a$, $a\in K$ and then $\overline\sigma$ maps all adjoined roots to different roots of their characteristic polynomial.
Can I please have an example of a $K$-automorphism of $L$, and how it permutes the adjoined root to a different adjoined root of its characteristic polynomial?