Central simple algebra

Summary

A central simple algebra over a field $K$ is a finite-dimensional associative algebra over $K$ that is a simple ring. Its center must be exactly $K$. By Wedderburn's theorem each one is a matrix algebra $M_n(D)$ over a central division algebra $D$. They are stable under tensor product over $K$, and their classes up to matrix size form the Brauer group of $K$. They are central in algebraic number theory, in Galois cohomology and in the arithmetic of quadratic forms.