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) $n$.