Classification of Clifford algebras

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional Clifford algebras for a nondegenerate quadratic form are completely classified as rings. In general, the Clifford algebra is either a central simple algebra or a direct sum of two copies of such an algebra. For Clifford algebras over real or complex field, this means that the Clifford algebra is isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two such algebras that are (non-canonically) isomorphic. The dimensions of the matrix algebra, and what division ring (R, C, H) can be determined by the dimension of the vector space and invariants of the quadratic form (its signature, over the reals).