Central simple algebra

ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the Weyl algebra

K,\partial_X /math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.)

For example, the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s C form a CSA over themselves, but not over the

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s R (the center of C is all of C, not just R). The

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...

s H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or '' Brauer equivalent'') if their division rings ''S'' and ''T'' are isomorphic. The set of all

equivalence clas