Central simple algebra

• 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

${\displaystyle \mathrm {ind} (D)=\prod _{i=1}^{r}p_{i}^{m_{i}}\ }$

then D has a tensor product decomposition

${\displaystyle D=\bigotimes _{i=1}^{r}D_{i}\ }$

where each component D_i is a central division algebra of index ${\displaystyle p_{i}^{m_{i}}}$, and the components are uniquely determined up to isomorphism.[11]

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

${\displaystyle t+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} \leftrightarrow \left({\begin{array}{*{20}c}t+xi&y+zi\\-y+zi&t-xi\end{array}}\right).}$

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 t² + x² + y² + z² and reduced trace 2t.

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.

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

• 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.