 Cube-class group of a field

The cube-class group of a field $K$ is the quotient abelian groupTwo nonzero elements $a,b$ determine the same class precisely when $a/b$ is a cube in $K$. Multiplication of representatives gives the quotient operation. Every class has order dividing three because the cube of its representative is trivial in the quotient.

For $K=\mathbb{R}$, every nonzero real number has a real cube root, so the group is trivial. For a finite field ${\mathbb{F}}_q$, cyclicity of ${\mathbb{F}}_q^{\times }$ gives a cube-class group of order $\gcd (3,q-1)$. In $\mathbb{Q}$, the class of two is nontrivial and has order three: prime exponents in a rational cube must be divisible by three.

When the characteristic is not three and $K$ contains the cube roots of unity, Kummer theory connects these classes with extensions formed by adjoining cube roots. The quotient group itself is defined without those additional Kummer hypotheses.