# Central simple algebra

In ring theory 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*. As an example, note that any simple algebra is a central simple algebra over its center.

For example, the complex numbers **C** form a CSA over themselves, but not over the real numbers **R** (the center of **C** is all of **C**, not just **R**). The quaternions **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 classes of central simple algebras over a given field *F*, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(*F*) of the field *F*.^{[1]} It is always a torsion group.^{[2]}

We call a field *E* a *splitting field* for *A* over *K* if *A*⊗*E* is isomorphic to a matrix ring over *E*. Every finite dimensional CSA has a splitting field: indeed, in the case when *A* is a division algebra, then a maximal subfield of *A* is a splitting field. In general by theorems of Wedderburn and Koethe there is a splitting field which is a separable extension of *K* of degree equal to the index of *A*, and this splitting field is isomorphic to a subfield of *A*.^{[12]}^{[13]} As an example, the field **C** splits the quaternion algebra **H** over **R** with

We can use the existence of the splitting field to define **reduced norm** and **reduced trace** for a CSA *A*.^{[14]} Map *A* to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively. For example, in the quaternion algebra **H**, the splitting above shows that the element *t* + *x* **i** + *y* **j** + *z* **k** has reduced norm *t*^{2} + *x*^{2} + *y*^{2} + *z*^{2} and reduced trace 2*t*.

The reduced norm is multiplicative and the reduced trace is additive. An element *a* of *A* is invertible if and only if its reduced norm in non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.^{[15]}

CSAs over a field *K* are a non-commutative analog to extension fields over *K* – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be a division algebra). This is of particular interest in noncommutative number theory as generalizations of number fields (extensions of the rationals **Q**); see noncommutative number field.