Kummer theory

Summary

Kummer theory describes the abelian extensions of exponent dividing $n$ of a field $K$ that contains the $n$-th roots of unity. Every such extension is obtained by adjoining $n$-th roots of elements of $K$, and these extensions correspond to subgroups of $K^\times$ containing $K^{\times n}$. The Galois group is dual to the corresponding group of radicals through the Kummer pairing. Kummer theory is a basic tool in class field theory and in Iwasawa theory.