Introduction

This chapter deals with number fields and orders there of. We follow the common terminology and conventions as e.g. used in [2], [1], [3] or [4].

If $K$ is a number field, then an order $\mathcal{O}$ of $K$ is a subring of the ring of integers ${\mathcal{O}}_K$ of $K$, which is free of rank $[K:\mathbf{Q}]$ as a $\mathbf{Z}$-module. Depending on whether $K$ is an absolute field or relative field, orders are treated differently. As far as possible, the interaction and the interface for orders of absolute number fields and of relative number fields is the same.

Orders of absolute number fields

Assume that $K$ is defined as an absolute field. An order $\mathcal{O}$ of such a field are constructed (implicitly) by specifying a $\mathbf{Z}$-basis, which is referred to as the basis of $\mathcal{O}$. If $({\omega }_1,\dots ,{\omega }_d)$ is the basis of $\mathcal{O}$ and $({\alpha }_1,\dots ,{\alpha }_d)$ the basis of $K$, then the matrix $B\in {\mathrm{Mat}}_{d\times d}(\mathbf{Q})$ with

$$\left(\begin{array}{c}{\omega }_1\\ ⋮\\ {\omega }_d\end{array}\right)=B\left(\begin{array}{c}{\alpha }_1\\ ⋮\\ {\alpha }_d\end{array}\right)$$

is the basis matrix of $K$. If $K=\mathbf{Q}(\alpha )=\mathbf{Q}[x]/(f)$ is simple with $f\in \mathbf{Z}[x]$, then natural order $\mathbf{Z}[\alpha ]=\mathbf{Z}[x]/(f)$ is called the equation order of $K$.

Orders of relative number fields

Orders in non-absolute number fields, that is, relative extensions, are represented differently. Let $L/K$ be a finite extension of number fields, then currently we require any order in $L$ to contain ${\mathcal{O}}_K$, the ring of integers of $K$. In this case, an order $\mathcal{O}$ in $L$ is a finitly generated torsion-free module over the Dedekind domain ${\mathcal{O}}_K$. As a ring, the order $\mathcal{O}$ is unitary and has $L$ as a fraction field. Due to ${\mathcal{O}}_K$ in general not being a principal ideal domain, the module structure is more complicated and requires so called pseudo-matrices. See here for details on pseudo-matrices, or [1], Chapter 1 for an introduction.

In short, $\mathcal{O}$ is represented as $\sum {\mathfrak{a}}_i{\omega }_i$ with fractional ${\mathcal{O}}_K$ ideals ${\mathfrak{a}}_i\subset K$ and $K$-linear independent elements ${\omega }_i\in L$. In general it is impossible to have both ${\mathfrak{a}}_i$ integral and ${\omega }_i\in \mathcal{O}$, thus coefficients will not be integral and/or generators not in the structure.