Automorphism scheme of a finite field extension

Abstract:

Let $k\to K$ be a finite field extension and let us consider the automorphism scheme $Aut_kK$. We prove that $Aut_kK$ is a complete $k$-group, i.e., it has trivial centre and any automorphism is inner, except for separable extensions of degree 2 or 6. As a consequence, we obtain for finite field extensions $K_1, K_2$ of $k$, not being separable of degree 2 or 6, the following equivalence: \begin{equation*} K_1\simeq K_2 \Leftrightarrow Aut_kK_1\simeq Aut_kK_2.\end{equation*}