Kummer theoryKummerTheory2005-02-22 10:07:212005-02-22 14:34:57TheoremKummer extensionKummerabelian extension% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}The following theorem is usually referred to as {\it Kummer theory}.
\begin{thm}[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:
\begin{enumerate}
\item The extension $K(\sqrt[n]{a})$ for $a\in K$ is a cyclic extension over $K$ of degree dividing $n$.\\
\item Any cyclic extension of degree $n$ over $K$ is of the form $K(\sqrt[n]{a})$ for some $a\in K$.\\
\end{enumerate}
\end{thm}
\begin{defn}
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 {\bf Kummer extension} of $K$. Notice that the Galois group of the extension is of \PMlinkname{exponent}{Exponent} $n$.
\end{defn}