algebraic torus

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