Classification theorem for separable algebras

Question

1. Context
My lecture notes propose the following structure theorem for separable algebras (which is reminiscent of the Artin-Wedderburn theorem):

Let $k$ be a field. A $k$-algebra is separable if and only if $A\cong \bigoplus\limits_{i=1}^r A_i$ is a direct sum of finite-dimensional, simple $k$-algebras where all $Z(A_i)/k$ are separable extensions of fields.

2.Question

• Where can I find a proof of this structure theorem? Alternatively, an outline of a proof would be appreciated as well.

Answer 1

I give a proof in this blog post although it could probably be simplified and apparently there's a small gap I never got around to plugging. Whoops.