central simple algebra
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[edit]Noun
[edit]central simple algebra (plural central simple algebras)
- (algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.
- The complex numbers form a central simple algebra over themselves, but not over the real numbers (the centre of is all of , not just ). The quaternions form a 4-dimensional central simple algebra over .
- The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).
- 1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products[1], Elsevier (North-Holland), page 151:
- This crossed product was introduced by Noether and played a significant role in the classical theory of central simple algebras.
- 2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:
- Because of Wedderburn's theorem it is natural to call two central-simple algebras similar if they are isomorphic to matrix algebras over the same division algebra .
- 2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:
- Let be central simple algebras over a field . Then can be shown to be a central simple algebra over . Further, if is a central simple algebra over a field , then . I.e., it is isomorphic to a matrix algebra.
Synonyms
[edit]- CSA (initialism)
Hypernyms
[edit]Translations
[edit]type of associative algebra over a field
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Further reading
[edit]- Azumaya algebra on Wikipedia.Wikipedia
- Brauer group on Wikipedia.Wikipedia
- Severi–Brauer variety on Wikipedia.Wikipedia
- Albert–Brauer–Hasse–Noether theorem on Wikipedia.Wikipedia
- Central simple algebra on Encyclopedia of Mathematics