Kummer theory

Let $n\ge 1$ be prime to the characteristic of a field $K$, and suppose $K$ contains all $n$th roots of unity. Kummer theory classifies finite abelian extensions of exponent dividing $n$ by finite subgroups of $K^{\times }/(K^{\times }{)}^n$: to a subgroup generated by classes $a_i$ associate $K(a_i^{1/n}:i)$. The pairing sending an automorphism and a radicand class to $\sigma (a^{1/n})/a^{1/n}$ identifies the Galois group with the character group of the radicand subgroup.

In particular, cyclic extensions of degree $n$ correspond to cyclic subgroups of order $n$, rather than to a unique choice of one generator. The cohomological form isFor $n=2$ in characteristic different from two, the nonzero square classes other than the identity classify quadratic extensions. The assumptions on roots of unity and characteristic ensure that the radicals form separable Galois extensions.^[ref]