Multiplicative group of a field

Definition as a group
Suppose $$K$$ is a field. The multiplicative group of $$K$$, denoted $$K^*$$ or $$K^\times$$ or $$\mathbb{G}_m(K)$$, is defined as the group of all nonzero elements of $$K$$ under multiplication.

Definition as an algebraic group
Suppose $$K$$ is a field. The multiplicative group of $$K$$, denoted $$K^*$$ or $$K^\times$$ or $$\mathbb{G}_m(K)$$, is defined as the following algebraic group:


 * 1) The underlying algebraic variety is the set of nonzero elements of $$K$$, which can be viewed as an open subset of $$K$$ or of $$\mathbb{P}^1(K)$$.
 * 2) The group structure is as the group of all nonzero elements of $$K$$ under multiplication.

Definition as a linear algebraic group
Suppose $$K$$ is a field. The multiplicative group of $$K$$ is the general linear group of degree one over $$K$$.

Terminology
The multiplicative group of a field is the unique one-dimensional algebraic torus over the field. It is also a split torus. An algebraic torus over a field is a direct product of multiplicative groups of field extensions. A split torus is a direct product of copies of the multiplicative group of the field.

Facts

 * Multiplicative group of a finite field is cyclic, (in particular, this is true for finite prime fields).
 * Multiplicative group of a field implies every finite subgroup is cyclic
 * Classification of connected one-dimensional algebraic groups over an algebraically closed field