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# orders in a number field

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 module in $K$. If the module contains as many over $\mathbb{Z}$ linearly independent elements as is the degree of $K$ over $\mathbb{Q}$, then the module is complete.

If a complete module in $K$ contains the unity 1 of $K$ and is a ring, it is called an order (in German: Ordnung) in the field $K$.

A number $\alpha$ of the algebraic number field $K$ is called a coefficient of the module $M$, if $\alpha M\subseteq M$.

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.

Theorem 2. If $\alpha$ belongs to an order in the field, then the coefficients of the characteristic equation 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 maximal order or the principal order (in German: Hauptordnung). The set of the orders is partially ordered by the set inclusion.

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 $\sqrt{2}$.

# References

- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).

## Mathematics Subject Classification

12F05*no label found*11R04

*no label found*06B10

*no label found*

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## Comments

## No HTML for number order in fields

TeX source shows, page images shows, but HTML with images is not showing on my computer. I'm using IE 6.0 on a Dell with Windows XP.

## Re: No HTML for number order in fields

The cause is not the IE (although it isn'n recommended for PM; Firefox and Opera are better). The cause seems to be the "hats" over the name Safarevic (used in the German translation of the Russian original book). Maybe there could exist a better way than \v{S} and \v{c} for forming these sibilants.