A homework problem is divided into 2 parts:
I managed to solve the first part, which states:
Prove Koethe's theorem: If $D$ is a finite dimensional central division $k$-algebra and $K_0 \subset D$ is a separable extension of $k$ then $D$ has a maximal subfield $K$ such that $K_0 \subset K$ and $K/k$ is separable.
I am having difficulties with the second part:
Deduce that if $S$ is a finite dimensional central simple $k$-algebra then there exists a Galois extension $K/k$ such that $K$ splits $S$.
My attempt at the second part:
The algebra $S$ is not necessarily a division algebra, but even if it was, Koethe's theorem would give me a maximal subfield separable over $k$. This subfield would split $S$ because it is maximal, but to show that it is Galois I would still need to prove that it is normal over $k$.
I think that one of the things I might be missing is the relation between a subfield that splits an algebra and the usual notion of a splitting field.