# 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).

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.

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.

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. It is always a torsion group.

We call a field E a splitting field for A if AE 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. As an example, the field C splits the quaternion algebra H over R with $t + x mathbf{i} + y mathbf{j} + z mathbf{k} leftrightarrow left({begin{array}{*{20}c} t + x i & y + z i \ -y + z i & t - x i end{array}}right) .$

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 element t + x i + y j + z k has reduced norm t2 + x2 + y2 + z2 and reduced trace 2t.

### Other articles related to "simple, algebra, central simple algebra, simple algebra":

Simple Past
... The simple past or past simple, sometimes called the preterite, is the basic form of the past tense in Modern English ... The term "simple" is used to distinguish the syntactical construction whose basic form uses the plain past tense alone, from other past tense constructions which ... Regular verbs form the simple past in -ed however there are a few hundred irregular verbs with different forms ...
Brauer Group and Class Field Theory
... of order n can be represented by a cyclic division algebra of dimension n2 ... If D is a central simple algebra over K and v is a valuation then D ⊗ Kv is a central simple algebra over Kv, the local completion of K at v ... A given central simple algebra D splits for all but finitely many v, so that the image of D under almost all such homomorphisms is 0 ...
Artin–Wedderburn Theorem - Examples
... Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H ... Every central simple algebra over R must be a matrix ring over R or H ... Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C ...
Central Simple Algebra - Properties
... There is a unique division algebra in each Brauer equivalence class ... 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 ...

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