Let {{tmath| \mathbb{R} }} be the field of real numbers, {{tmath| \mathbb{C} }} be the field of complex numbers, and {{tmath| \mathbb{H} }} the
quaternions. • A
central simple algebra (sometimes called a
Brauer algebra) is a simple finite-dimensional algebra over a
field whose
center is . • Every finite-dimensional simple algebra over {{tmath| \mathbb{R} }} is isomorphic to an algebra of matrices with entries in {{tmath| \mathbb{R} }}, {{tmath| \mathbb{C} }}, or {{tmath| \mathbb{H} }}. Every
central simple algebra over {{tmath| \mathbb{R} }} is isomorphic to an algebra of matrices with entries {{tmath| \mathbb{R} }} or {{tmath| \mathbb{H} }}. These results follow from the
Frobenius theorem. • Every finite-dimensional simple algebra over {{tmath| \mathbb{C} }} is a central simple algebra, and is isomorphic to a matrix ring over {{tmath| \mathbb{C} }}. • Every finite-dimensional central simple algebra over a
finite field is isomorphic to a matrix ring over that field. • Over a field of characteristic zero, the
Weyl algebra is simple but not semisimple, and in particular not a matrix algebra over a division algebra over its center; the Weyl algebra is infinite-dimensional, so Wedderburn's theorem does not apply to it. == See also ==