---
tag: XZQENYV15Y
kind: concept
name: Kummer theory
msc: [11R20, 12F10]
sources:
  - url: https://en.wikipedia.org/wiki/Kummer_theory
same_as:
  wikidata: Q1548483
  wikipedia: 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|0YV2CKP2AV]].
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|BEKCEYHCHX]] and in [[Iwasawa theory|Y2SPPSQ5D4]].

## Explanations

### L1 human

Let $K$ be a field of characteristic other than $2$, so that $\pm 1$ are two distinct square roots of unity.
Then every extension of $K$ of degree $2$ is obtained by adjoining a square root, as with $\Q(\sqrt{2})$ over $\Q$.
Kummer theory is the same idea for $n$-th roots.
Suppose that $K$ contains $n$ distinct $n$-th roots of unity.
Then adjoining an $n$-th root $\sqrt[n]{a}$ gives a cyclic extension.
All abelian extensions whose Galois group has exponent dividing $n$ arise by adjoining several such roots.
For example, $\Q(i)$ contains the fourth roots of unity, and $\Q(i, \sqrt[4]{2})$ is a cyclic extension of $\Q(i)$ of degree $4$.