Splitting field of a central simple algebra

Summary

Let $A$ be a central simple algebra over a field $K$. A field extension $L$ of $K$ splits $A$ if $A$ becomes a full matrix algebra after extending scalars to $L$. Every central simple algebra has a splitting field of finite degree over $K$, and even one that is a finite Galois extension of $K$. Splitting fields link central simple algebras with Galois cohomology: the algebras split by $L$ give the relative Brauer group $\mathrm{Br}(L/K)$. The word is unrelated to the splitting field of a polynomial, apart from the idea of making something decompose.