abelian number fieldAbelianNumberField2006-06-20 13:35:182006-07-19 09:50:38Definitionnumber fieldcyclic number field% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}\begin{defn}
An abelian number field is a number field $K$ such that $K/\Rats$ is an abelian extension, i.e. $K/\Rats$ is Galois and $\Gal(K/\Rats)$ is an abelian group.
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The abelian number fields are classified by the Kronecker-Weber Theorem.
\begin{defn}
A cyclic number field is an (abelian) number field $K$ such that $K/\Rats$ is a Galois extension and $\Gal(K/\Rats)$ is a finite cyclic group (therefore abelian).
\end{defn}