Brauer group

The quaternions are "essentially" the alone (non-trivial) axial simple algebra (CSA) over the absolute numbers, in the faculty that every CSA over the reals is Brauer agnate to either the reals or the quaternions. Explicitly, the Brauer accumulation of the reals consists of two classes, represented by the reals and the quaternions, area the Brauer accumulation is the set of all CSAs, up to adequation affiliation of one CSA getting a cast ring over another. By the Artin–Wedderburn assumption (specifically, Wedderburn's part), CSAs are all cast algebras over a analysis algebra, and appropriately the quaternions are the alone non-trivial analysis algebra over the reals.

CSAs – rings over a field, which are simple algebras (have no non-trivial 2-sided ideals, just as with fields) whose centermost is absolutely the acreage – are a noncommutative analog of addendum fields, and are added akin than accepted ring extensions. The actuality that the quaternions are the alone non-trivial CSA over the reals (up to equivalence) may be compared with the actuality that the circuitous numbers are the alone non-trivial acreage addendum of the reals.