In ring theory and related areas of mathematics a central simple algebra over a fieldK is a finite-dimensional associative algebraA, 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 numbersR. 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. Given two central simple algebras A ~ M and B ~ M over the same field F, A and B are called similar 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 of the fieldF. It is always a torsion group.
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. The Schur index of a central simple algebra is the degree of the equivalent division algebra: it depends only on the Brauer class of the algebra.
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, and the two numbers are composed of the same prime factors.
If S is a simple subalgebra of a central simple algebra A then dimFS divides dimFA.
Every 4-dimensional central simple algebra over a fieldF 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
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. 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. 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 + xi + yj + zk 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.
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. This is of particular interest in noncommutative number theory as generalizations of number fields ; see noncommutative number field.
