Automorphism groups of finite dimensional simple algebras

Abstract

We show that if a field $k$ contains sufficiently many elements (for instance, if $k$ is infinite), and $K$ is an algebraically closed field containing $k$, then every linear algebraic $k$-group over $K$ is $k$-isomorphic to $\mathop{\rm Aut}(A\otimes_k\!K)$, where $A$ is a finite dimensional simple algebra over $k$.