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:
$$K(\sqrt[n]{{a}_{1}},\sqrt[n]{{a}_{2}},\mathrm{\dots},\sqrt[n]{{a}_{k}})$$ 
with ${a}_{i}\mathrm{\in}{K}^{\mathrm{\times}}$ 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 

Canonical name  KummerTheory 
Date of creation  20130322 15:04:20 
Last modified on  20130322 15:04:20 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  5 
Author  alozano (2414) 
Entry type  Theorem 
Classification  msc 12F05 
Related topic  AbelianExtension 
Related topic  CyclicExtension 
Related topic  Exponent 
Defines  Kummer extension 