# orders in a number field

If ${\mu}_{1},\mathrm{\dots},{\mu}_{m}$ are elements of an algebraic number field^{} $K$, then the subset

$$M=\{{n}_{1}{\mu}_{1}+\mathrm{\dots}+{n}_{m}{\mu}_{m}\in K\mathrm{\vdots}{n}_{i}\in \mathbb{Z}\forall i\}$$ |

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 (http://planetmath.org/NumberField) of $K$ over $\mathbb{Q}$, then the module is complete.

If a complete module in $K$ 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^{} (http://planetmath.org/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 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).

Title | orders in a number field |

Canonical name | OrdersInANumberField |

Date of creation | 2013-03-22 16:52:46 |

Last modified on | 2013-03-22 16:52:46 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 17 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 12F05 |

Classification | msc 11R04 |

Classification | msc 06B10 |

Related topic | Module |

Defines | module |

Defines | complete |

Defines | order of a number field |

Defines | principal order |

Defines | maximal order |