Multiplicative group of a field

Definition

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.

Arithmetic functions

Function	Value	Similar groups	Explanation
dimension of an algebraic group	1	classification of connected one-dimensional groups over an algebraically closed field	as an algebraic variety, it can be viewed as an open subset of $K$, which is one-dimensional.
rank of an algebraic group	1		it is nilpotent, so its rank equals its dimension.

Algebraic group properties

Property	Satisfied?	Explanation
connected algebraic group	Yes if $K$ is infinite No if $K$ is finite	as an algebraic variety, it is a cofinite subset of $K^{1}$. The induced topology on it is the cofinite topology, i.e., every nonempty open subset is cofinite. Thus, it is connected for infinite $K$. Note that the group need not be connected under topologies arising from analytic structures on $K$, i.e., it need not be a connected Lie group. For instance, over $\mathbb {R}$, it is not connected. See connected algebraic group need not be connected as a Lie group.
abelian algebraic group	Yes	By definition, multiplication in a field is commutative
unipotent algebraic group	No unless $K$ is field:F2	With the standard view as $GL(1,K)$, the only unipotent element is the identity element.

Subgroup structure

Further information: subgroup structure of multiplicative group of a field

Easy facts

• Multiplicative group of a field implies every finite subgroup is cyclic
• Multiplicative group of a finite field is cyclic, and in particular multiplicative group of a prime field is cyclic

Closed normal subgroups and quotients

For a finite field, the group is cyclic of order one less than the size of the field, and the topology is discrete, so the study proceeds just like the study of the subgroup structure of finite cyclic groups.

For an infinite field, the closed normal subgroups are precisely the finite subgroups and the whole group. Further, the finite subgroups are precisely the subgroups of $n^{th}$ roots of unity for positive integers $n$. Any such subgroup is cyclic of order dividing $n$ (and equal to $n$ if a primitive $n^{th}$ root exists, which is always the case for an algebraically closed field whose characteristic does not divide $n$).

The quotient group by this subgroup is isomorphic to the subgroup of the multiplicative group comprising all $n^{th}$ powers. This is because the closed normal subgroup of $n^{th}$ roots of unity is an endomorphism kernel corresponding to the endomorphism $x\mapsto x^{n}$. In particular, when the field is an algebraically closed field, the endomorphism is a surjective endomorphism and thus the quotient by the subgroup of $n^{th}$ roots of unity is isomorphic to the multiplicative group itself.

Algebraic defining functions

Subgroup-defining functions

Subgroup-defining function	Meaning	Value
unipotent radical of an algebraic group	largest connected closed normal unipotent subgroup	trivial group
radical of an algebraic group	largest connected closed normal solvable subgroup	whole group

Conjugacy class-defining functions

Conjugacy class-defining function	Meaning	Value
Borel subgroup		whole group
Cartan subgroup		whole group

Associated constructs

Associated construct	Value
formal group law of an algebraic group, becomes a formal group law over the same field	multiplicative formal group law
Lie algebra of an algebraic group, becomes a Lie algebra over the same field	becomes a one-dimensional abelian Lie algebra

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