Automorphism scheme of a finite field extension

Abstract

Let k → K k\to K be a finite field extension and let us consider the automorphism scheme A u t k K Aut_kK . We prove that A u t k K Aut_kK is a complete k 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 K_1, K_2 of k k , not being separable of degree 2 or 6, the following equivalence: K 1 ≃ K 2 ⇔ A u t k K 1 ≃ A u t k K 2 . \begin{equation*} K_1\simeq K_2 \Leftrightarrow Aut_kK_1\simeq Aut_kK_2.\end{equation*}