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# 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*

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