# 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. 1.

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

2. 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 (http://planetmath.org/Exponent) $n$.

Title Kummer theory KummerTheory 2013-03-22 15:04:20 2013-03-22 15:04:20 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 12F05 AbelianExtension CyclicExtension Exponent Kummer extension