In mathematics, an algebraic torus over a field K is an algebraic group which is isomorphic over the algebraic closure of K to Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Jump to: navigation, search In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

(GL₁)^r

for some integer r, the rank of the torus. Here, GL₁ = G_m is the multiplicative algebraic group. Tori are therefore always commutative. If this isomorphism can be realised over K itself, then the torus is said to be split. These groups were named by analogy with the theory of tori in Lie group theory (see maximal torus). Rank means a wide variety of things in mathematics, including: Rank (linear algebra) Rank of a tensor Rank of an array Rank of an abelian group Rank (set theory) Rank-into-rank Rank of a greedoid This is a disambiguation page — a navigational aid which lists other pages that might... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... Jump to: navigation, search In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. ...

Examples of non-split tori can be constructed by means of Weil restriction; in fact, in general, every isomorphism class of tori contains a torus which is a product of Weil restrictions of split tori. Each algebraic torus is dual (as an Abelian group) to a Galois module, its set of algebraic group homomorphisms to GL₁. (These statements are true for perfect fields. For non-perfect fields, they should be qualified to take account of inseparability questions.) In mathematics, specifically the theory of algebraic groups, Weil restriction is a functor allowing one to pass from an algebraic group G over a field L to another one, RG, over a subfield K. The idea is that the group of points G(L) of G over L should be... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense... // Homomorphism for beginners Homomorphism is one of the fundamental concepts in abstract algebra. ... In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable... In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable...

See also:

• Torus based cryptography
• Torus embedding.