# 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 algebra *A*, which is simple, and for which the center is exactly *K*. In other words, 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]}

## Properties

- According to the Artin–Wedderburn theorem a finite-dimensional simple algebra
*A*is isomorphic to the matrix algebra*M*(*n*,*S*) for some division ring*S*. Hence, there is a unique division algebra in each Brauer equivalence class.^{[3]} - Every automorphism of a central simple algebra is an inner automorphism (follows from Skolem–Noether theorem).
- The dimension of a central simple algebra as a vector space over its centre is always a square: the
**degree**is the square root of this dimension.^{[4]}The**Schur index**of a central simple algebra is the degree of the equivalent division algebra:^{[5]}it depends only on the Brauer class of the algebra.^{[6]} - The
**period**or**exponent**of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index,^{[7]}and the two numbers are composed of the same prime factors.^{[8]}^{[9]}^{[10]} - If
*S*is a simple subalgebra of a central simple algebra*A*then dim_{F}*S*divides dim_{F}*A*. - Every 4-dimensional central simple algebra over a field
*F*is isomorphic to a quaternion algebra; in fact, it is either a two-by-two matrix algebra, or a division algebra. - If
*D*is a central division algebra over*K*for which the index has prime factorisation

- then
*D*has a tensor product decomposition - where each component
*D*_{i}is a central division algebra of index , and the components are uniquely determined up to isomorphism.^{[11]}

## Splitting field

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]}

## Generalization

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.

## See also

- Azumaya algebra, generalization of CSAs where the base field is replaced by a commutative local ring
- Severi–Brauer variety
- Posner's theorem

## References

- ↑ Lorenz (2008) p.159
- ↑ Lorenz (2008) p.194
- ↑ Lorenz (2008) p.160
- ↑ Gille & Szamuely (2006) p.21
- ↑ Lorenz (2008) p.163
- ↑ Gille & Szamuely (2006) p.100
- ↑ Jacobson (1996) p.60
- ↑ Jacobson (1996) p.61
- ↑ Gille & Szamuely (2006) p.104
- ↑ Cohn, Paul M. (2003).
*Further Algebra and Applications*. Springer-Verlag. p. 208. ISBN 1852336676. - ↑ Gille & Szamuely (2006) p.105
- ↑ Jacobson (1996) pp.27-28
- ↑ Gille & Szamuely (2006) p.101
- ↑ Gille & Szamuely (2006) pp.37-38
- ↑ Gille & Szamuely (2006) p.38

- Cohn, P.M. (2003).
*Further Algebra and Applications*(2nd ed.). Springer. ISBN 1852336676. Zbl 1006.00001. - Jacobson, Nathan (1996).
*Finite-dimensional division algebras over fields*. Berlin: Springer-Verlag. ISBN 3-540-57029-2. Zbl 0874.16002. - Lam, Tsit-Yuen (2005).
*Introduction to Quadratic Forms over Fields*. Graduate Studies in Mathematics.**67**. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. - Lorenz, Falko (2008).
*Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics*. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.

### Further reading

- Albert, A.A. (1939).
*Structure of Algebras*. Colloquium Publications.**24**(7th revised reprint ed.). American Mathematical Society. ISBN 0-8218-1024-3. Zbl 0023.19901. - Gille, Philippe; Szamuely, Tamás (2006).
*Central simple algebras and Galois cohomology*. Cambridge Studies in Advanced Mathematics.**101**. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.