by the equation in the affine plane over with coordinates . The multiplication is then given by restricting the regular rational map defined by and the inverse is the restriction of the regular rational map .

Definition

Let be a field with algebraic closure . Then a -torus is an algebraic group defined over which is isomorphic over to a finite product of copies of the multiplicative group.

In other words, if is an -group it is a torus if and only if for some . The basic

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