Quaternion algebra

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F^[1][2] that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, ${\displaystyle A\otimes _{F}K}$ is isomorphic to the 2 × 2 matrix algebra over K.

The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over ${\displaystyle F=\mathbb {R} }$, and indeed the only one over ${\displaystyle \mathbb {R} }$ apart from the 2 × 2 real matrix algebra, up to isomorphism. When ${\displaystyle F=\mathbb {C} }$, then the biquaternions form the quaternion algebra over F.