ALGEBRAIC TORUS

In mathematics, an 'algebraic torus' over a field ''K'' is an algebraic group which is isomorphic over the algebraic closure of ''K'' to
:''(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).
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.)
See also:

★ Torus based cryptography

★ Torus embedding.