central simple algebra

Fix a base field K. Among all the finite-dimensional algebras over K, the central simple ones are those that are as tightly fastened to K as possible: they have no nontrivial two-sided ideals, and the only elements commuting with everything are the scalars from K itself. There is no hidden bigger field of constants inside, and no way to split off an ideal. They are the irreducible atoms of algebra over K.

Precisely, a central simple algebra over K is a finite-dimensional K-algebra A that is simple (only ideals 0 and A) and whose center Z(A) equals K. By Wedderburn each such A is a matrix ring M_n(D) for a division algebra D with center K; conversely any such matrix algebra is central simple. A clean intrinsic test: A is central simple over K if and only if A tensor with the algebraic closure of K becomes a full matrix algebra M_m(K-bar).

Central simple algebras carry a rich arithmetic. Their dimension over K is always a perfect square n^2, so one speaks of the degree n. The tensor product of two central simple K-algebras is again central simple, and A tensor A-op is isomorphic to a full matrix algebra over K. These facts organize central simple algebras into the Brauer group, the central object of the theory.

The reason they matter beyond ring theory: over local and global fields they are classified by Galois cohomology, and this is the engine of class field theory. A quaternion algebra is the smallest nontrivial case, degree 2, and already encodes deep arithmetic — for instance whether a conic has a rational point.