|
|
|
Viewing Version
1
of
'Hermite's Theorem'
|
[ view 'Hermite's Theorem'
|
back to history
]
| Title of object: |
Hermite's Theorem |
| Canonical Name: |
HermitesTheorem |
| Type: |
Corollary |
| Created on: |
2005-02-24 08:35:07 |
| Modified on: |
2005-02-24 08:35:07 |
| Classification: |
msc:11H06, msc:11R29 |
| Defines: |
unramified outside a set of primes |
Revision comment (for changes between this and next version):
| Changes for correction #10207 ('title'). |
Preamble:
% 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}
% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}} |
Content:
The following is a corollary of Minkowski's theorem on ideal classes, which is a corollary of Minkowski's theorem on lattices.
\begin{defn}
Let $S=\{p_1,\ldots,p_r\}$ be a set of rational primes $p_i \in \Ints$. We say that a number field $K$ is {\bf unramified outside $S$} if any prime not in $S$ is unramified in $K$. In other words, if $p$ is ramified in $K$, then $p\in S$. In other words, the only primes that divide the discriminant of $K$ are elements of $S$.
\end{defn}
\begin{cor}[Hermite's Theorem]
Let $S=\{p_1,\ldots,p_r\}$ be a set of rational primes $p_i \in \Ints$ and let $N\in \Nats$ be arbitrary. There is only a finite number of fields $K$ which are unramified outside $S$ and bounded degree $[K:\Rats]\leq N$.
\end{cor} |
|
|
|
|
|
