Classification of separable algebras up to Morita equivalence Is there a simple classification of separable algebras up to Morita equivalence, working over a particular field $k$?
For example, over $\mathbb{C}$, every separable algebra is Morita equivalent to the commutative algebra $\mathbb{C}^n$ for some $n$. Is something similar true in general?
Edit: It seems this is hard in general. So let me ask something a bit different: what property of a field $k$ implies that all separable algebras over it are Morita equivalent to $k^n$?
 A: A separable algebra over a field $k$ is a finite product
$$M_{n_1}(D_1)\times\dots\times M_{n_r}(D_r)$$
of matrix rings over division algebras $D_i$ whose centres are finite separable field extensions of $k$, and two such algebras are Morita equivalent if and only if the sequences of division algebras $D_1,\dots,D_r$ are the same up to a permutation and isomorphisms (of division algebras over $k$ if by Morita equivalence you mean Morita equivalence "over $k$", or of abstract rings if you mean Morita equivalence of abstract rings).
If you mean Morita equivalence "over $k$", then the answer to the question in the edit is that $k$ should be separably closed. If not, then any separable field extension is a separable $k$-algebra that is not isomorphic to a product of copies of $k$. If $k$ is separably closed then its Brauer group is trivial, so every $D_i$ must be isomorphic to $k$.
If you mean Morita equivalence as abstract rings, then you also have to consider the possibility that $k$ is not separably closed, but every finite separable extension is isomorphic to $k$ as an abstract field (is that possible?).
