Brauer expounded central division algebras over a perfect field while in K�nigsberg; the isomorphism classes of such algebras form the elements of the Brauer group he introduced.
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Second, one can replace the complex numbers by any ( real ) division algebra, including ( for " n " = 1 ) the octonions.
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Wedderburn's little theorem states that if " D " is a finite division algebra, then " D " is a finite field.
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Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite dimensional central division algebras.
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Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite dimensional central division algebras.
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His main interests are division algebras, Gelfand Kirillov dimension, Coxeter groups, Artin groups, combinatorial group theory, monomial algebras, and arithmetic of algebraic groups.
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In pure mathematics, quaternions show up significantly as one of the four finite-dimensional normed division algebras over the real numbers, with applications throughout algebra and geometry.
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Equivalently, any division algebra of period dividing " n " is Brauer equivalent to a tensor product of cyclic algebras of degree " n ".
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Then Schur's lemma says that the endomorphism ring of the module " M " is a division algebra over the field " k ".
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Basic examples are symplectic groups but it is possible to construct more using division algebras ( for example the unit group of a quaternion algebra is a classical group ).
