Algebraic torus

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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.