11-00-AbelianNumberField.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{AbelianNumberField}
\pmcreated{2013-03-22 16:01:24}
\pmmodified{2013-03-22 16:01:24}
\pmowner{alozano}{2414}
\pmmodifier{alozano}{2414}
\pmtitle{abelian number field}
\pmrecord{5}{38063}
\pmprivacy{1}
\pmauthor{alozano}{2414}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{11-00}
\pmrelated{GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ}
\pmdefines{cyclic number field}

\endmetadata

% this is the default PlanetMath preamble.  as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}

% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}

\theoremstyle{definition}
\newtheorem{exa}{Example}

% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}}
\newcommand{\Gal}{\operatorname{Gal}}
\newcommand{\Cl}{\operatorname{Cl}}
\begin{document}
\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.
\end{defn}

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}
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\end{document}