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.

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

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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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