Editing Quaternion (section)

== Brauer group ==
{{further|Brauer group}}

The quaternions are "essentially" the only (non-trivial) [[central simple algebra]] (CSA) over the real numbers, in the sense that every CSA over the real numbers is [[Brauer equivalent]] to either the real numbers or the quaternions. Explicitly, the [[Brauer group]] of the real numbers consists of two classes, represented by the real numbers and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a [[matrix ring]] over another. By the [[Artin–Wedderburn theorem]] (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the real numbers.

CSAs – finite dimensional rings over a field, which are [[simple algebra]]s (have no non-trivial 2-sided ideals, just as with fields) whose center is exactly the field – are a noncommutative analog of [[extension field]]s, and are more restrictive than general ring extensions. The fact that the quaternions are the only non-trivial CSA over the real numbers (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial finite field extension of the real numbers.