orders in a number fieldOrdersInANumberField2007-03-30 11:59:522008-02-24 13:18:38Topicorder in an algebramodulecompleteorder of a number fieldprincipal ordermaximal orderorder% this is the default PlanetMath preamble. as your knowledge
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If\, $\mu_1,\,\ldots,\,\mu_m$\, are elements of an algebraic number field $K$, then the subset
$$M =
\{n_1\mu_1+\ldots+n_m\mu_m\in K\,\vdots\;\; n_i\in\mathbb{Z}\;\;\forall i\}$$
of $K$ is a $\mathbb{Z}$-module, called a {\em module in} $K$.\, If the module contains as many over $\mathbb{Z}$ linearly independent elements as is the \PMlinkname{degree}{NumberField} of $K$ over $\mathbb{Q}$, then the module is {\em complete}.
If a complete module in $K$ \PMlinkescapetext{contains} the unity 1 of $K$ and is a ring, it is called an {\em order} (in German: {\em Ordnung}) in the field $K$.\\
A number $\alpha$ of the algebraic number field $K$ is called a {\em coefficient of the module} $M$, if\, $\alpha M \subseteq M$.\,
\textbf{Theorem 1.}\; The set $\mathcal{L}_M$ of all coefficients of a complete module $M$ is an order in the field.\, Conversely, every order $\mathcal{L}$ in the number field $K$ is a coefficient ring of some module.
\textbf{Theorem 2.}\; If $\alpha$ belongs to an order in the field, then the coefficients of the \PMlinkname{characteristic equation}{CharacteristicEquation} of $\alpha$ and thus the coefficients of the minimal polynomial of $\alpha$ are rational integers.
Theorem 2 means that any order is contained in the ring of integers of the algebraic number field $K$.\, Thus this ring $\mathcal{O}_K$, being itself an order, is the greatest order; $\mathcal{O}_K$ is called the {\em maximal order} or the {\em principal order} (in German: {\em Hauptordnung}).\, The set of the orders is partially ordered by the set inclusion.
\textbf{Example.}\, In the field $\mathbb{Q}(\sqrt{2})$, the coefficient ring of the module $M$ generated by $2$ and $\frac{\sqrt{2}}{2}$ is the module $\mathcal{L}_M$ generated by $1$ and $2\sqrt{2}$.\, The maximal order of the field is generated by $1$ and $\sqrt{2}$.
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\bibitem{BS}{\sc S. Borewicz \& I. Safarevic}: {\em Zahlentheorie}.\, Birkh\"auser Verlag. Basel und Stuttgart (1966).
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