Let $k$ be a field. Then $k^*$ , the multiplicative group of $k$ is an affine algebraic group over $k$ . An affine algebraic group of the form $(k^*)^n$ is called an algebraic torus over $k$ .

The name is connected to the fact that if $k=\C$ , then an algebraic torus is the complexification of the standard torus $(S^1)^n$ .