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'the ring of integers of a number field is finitely generated over $\mathbb{Z}$'
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| Title of object: |
the ring of integers of a number field is finitely generated over $\mathbb{Z}$ |
| Canonical Name: |
RingOfIntegersOfANumberFieldIsFinitelyGeneratedOverMathbbZ |
| Type: |
Theorem |
| Created on: |
2005-03-17 15:25:43 |
| Modified on: |
2005-03-17 15:30:30 |
| Classification: |
msc:13B22 |
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}
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%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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\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}} |
Content:
\begin{thm}
Let $K$ be a number field of degree $n$ over $\Rats$ and let $\mathcal{O}_K$ be the ring of integers of $K$. The ring $\mathcal{O}_K$ is a free abelian group of rank $n$. In other words, there exists a finite integral basis (with $n$ elements) for $K$, i.e. there exist algebraic integers $\alpha_1,\ldots,\ \alpha_n$ such that every element of $\mathcal{O}_K$ can be expressed uniquely as a $\Ints$-linear combination of the $\alpha_i$.
\end{thm}
\begin{cor}
Every ideal of $\mathcal{O}_K$ is finitely generated.
\end{cor}
\begin{proof}[Proof of the corollary]
By the theorem, $\mathcal{O}_K$ is a free abelian group of rank $n$, and therefore it is finitely generated. Notice that an ideal is an additive subgroup. Finally a subgroup of a finitely generated free abelian group is also finitely generated.
\end{proof} |
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