## You are here

HomeKummer theory

## Primary tabs

# Kummer theory

The following theorem is usually referred to as Kummer theory.

###### Theorem 1 (Kummer Theory).

Let $n$ be a positive integer and let $K$ be a field of characteristic not dividing $n$ which contains the $n$-th roots of unity. Then:

1. The extension $K(\sqrt[n]{a})$ for $a\in K$ is a cyclic extension over $K$ of degree dividing $n$.

2. Any cyclic extension of degree $n$ over $K$ is of the form $K(\sqrt[n]{a})$ for some $a\in K$.

###### Definition 1.

Let $n$ be a positive integer and let $K$ be a field of characteristic not dividing $n$ which contains the $n$-th roots of unity. An extension of $K$ of the form:

$K(\sqrt[n]{a_{1}},\sqrt[n]{a_{2}},\ldots,\sqrt[n]{a_{k}})$ |

with $a_{i}\in K^{\times}$ is called a Kummer extension of $K$. Notice that the Galois group of the extension is of exponent $n$.

Defines:

Kummer extension

Keywords:

Kummer, abelian extension

Related:

AbelianExtension, CyclicExtension, Exponent

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

12F05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff