On the normal basis theorem

Abstract

The normal basis theorem is a fundamental result in Galois theory. For infinite fields, textbooks and monographs usually refer to a proof given by Artin in 1948. For finite fields, a completely different argument is commonly used. We give two short proofs of the normal basis theorem which work without this distinction.They build on Dedekind’s theorem on the linear independence of Galois automorphisms, and on the Krull–Schmidt theorem. The rest is elementary linear algebra. Both proofs are inspired by but simpler than the one given by Deuring in 1932.