= Cube-class group of a field
{title2=$K^\times/(K^\times)^3$}
The cube-class group of a <field> $K$ is the quotient <abelian group>
$$
K^\times/(K^\times)^3.
$$
Two 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.
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