division algebra in a sentence

Examples

1. 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.
2. Second, one can replace the complex numbers by any ( real ) division algebra, including ( for " n " = 1 ) the octonions.
3. Wedderburn's little theorem states that if " D " is a finite division algebra, then " D " is a finite field.
4. 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.
5. 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.
6. His main interests are division algebras, Gelfand Kirillov dimension, Coxeter groups, Artin groups, combinatorial group theory, monomial algebras, and arithmetic of algebraic groups.
7. 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.
8. Equivalently, any division algebra of period dividing " n " is Brauer equivalent to a tensor product of cyclic algebras of degree " n ".
9. Then Schur's lemma says that the endomorphism ring of the module " M " is a division algebra over the field " k ".
10. 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 ).