Algebraic torus

In mathematics, an algebraic torus is a particular kind of algebraic group, that becomes of the simple form

Π GL₁

of a direct product of finitely many copies of the multiplicative group GL₁, once the underlying field is extended to an algebraically closed field. Such groups are therefore always commutative. They were so named by analogy with the theory of tori in Lie group theory (see maximal torus).

Over a field K that is not algebraically closed there exist algebraic tori that are not isomorphic over K to such a product (called a split torus). Examples can be constructed by means of Weil restriction; in fact in general products of Weil restrictions construct all the isomorphism classes. Each algebraic torus is dual to a Galois module, its algebraic group homomorphisms to GL₁ as abelian group. (These statements are true for perfect fields, and should be qualified to take account of inseparability questions.)

See also:

• Torus based cryptography
• Torus embedding.