köä

K× denotes the multiplicative group of a field K. It consists of all nonzero elements of K, with the operation of multiplication. The group axioms hold: closure under multiplication, associativity, the identity element 1, and inverses for every nonzero element. Since fields are commutative rings with unity, K× is an abelian (commutative) group.

The elements of K× that have finite order are called roots of unity contained in K; these form the torsion subgroup of K×. In many fields this torsion is small or trivial (for example, only ±1 in the rational numbers).

Finite fields provide a standard concrete case: if K is a finite field with q elements, then K× is cyclic of order q−1. Consequently every nonzero element of a finite field is a power of any chosen primitive element.

Common examples:

- K = Q (the rational numbers): Q× is the set of all nonzero rationals under multiplication; it is infinite and not cyclic, and it is not finitely generated.

- K = R (the real numbers): R× consists of positive and negative real numbers and is isomorphic to {±1} × (0, ∞). It has two connected components.

- K = C (the complex numbers): C× is the set of nonzero complex numbers and is isomorphic, via polar form, to R × S^1, reflecting a radial part and a circular (rotation) part.

Related concepts include the additive group of K, and, in number theory and algebraic geometry, the structure of K× as a foundational object in the study of units, valuations, and field extensions.