A central simple algebra over a field is a finite-dimensional associative -algebra that is a simple ring and whose center is exactly .
Every finite-dimensional semisimple algebra is a finite product of matrix algebras over division algebras. In particular, every central simple algebra over is isomorphic to for a finite-dimensional central division algebra over . The integer and the isomorphism class of are unique.
The tensor product over of two central simple -algebras is central simple. After extending scalars to an algebraic closure, both factors and their tensor product become full matrix algebras; faithful flatness then descends simplicity and the center.
The Brauer group of a field consists of Morita equivalence classes of central simple -algebras. Multiplication is induced by tensor product, the identity is , and the inverse of is the class of the opposite algebra .
For a finite Galois extension , Galois group , and normalized two-cocycle , the crossed-product algebra iswith and . It is a central simple -algebra split by .
Cohomologous normalized two-cocycles define isomorphic crossed-product algebras: replacing by the coboundary associated with a one-cochain rescales the basis elements . Consequently the crossed-product construction induces a map
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Central simple algebras are a fundamental concept in algebra, particularly in the study of algebraic structures over fields. Let's break down what central simple algebras are: 1. **Algebra**: In the context of central simple algebras, an algebra refers to a vector space equipped with a multiplication operation that is associative and distributes over vector addition.