Recall that an finite-dimensional algebra $A$ over a field $k$ is central simple iff there is an iso

$A \otimes_k A^{op} \cong M_n(k)$

where $A^{op}$ is the opposite ring and $M_n(k)$ is the matrix ring.

On the other hand, a finite field extension $K / k$ is Galois iff there is an iso

$K \otimes_k K \cong K^{Aut_k(K)}.$

These two statements strike me as similar and I am wondering:

- Is there a Galois theory for non-commutative extensions in which the central simple algebras take the role of the Galois extensions in the usual Galois theory? Or, is there a theory which subsumes these two facts as some special cases?
- Is there some sort of absolute "non-commutative" Galois group which features automorphisms of all non-commutative extensions of $k$ at a time? I could imagine this group $Gal_{nc}(k)$ to map (on?)to the usual group $Gal(k)$.

Thank you!