Quaternion Algebra

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars (=tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2×2 matrix algebra over K.

The notion of a quaternion algebra can be seen as a generalization of the Hamilton quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over (the real number field), and indeed the only one over ℝ apart from the 2×2 real matrix algebra, up to isomorphism.