## Introduction

This chapter deals with absolute number fields and orders there of.

### Definitions and vocabulary

We begin by collecting the necessary definitions and vocabulary. This is in particular important for everything related to embeddings of number fields into archimdean fields, since they are at least two (slighlty) different normalizations.

#### Number fields

By an absolute number field we mean finite extensions of , which is of type `AnticNumberField`

and whose elements are of type `nf_elem`

. Such an absolute number field is always given in the form , where is an irreducible polynomial. We call , where is the degree the *power basis* of . If is any element of , then the *representation matrix* of is the matrix representing with respect to the power basis, that is,

Let be the signature of , that is, has real embeddings , , and complex embeddings , . In Hecke the complex embeddings are always ordered such that for . The -linear function is called the *Minkowski map* (or *Minkowski embedding*).

#### Orders

If is an absolute number field, then an *order* of is a subring of the ring of integers , which is free of rank as a -module. The natural order is called the *equation order* of . In Hecke orders of absolute number fields are constructed (implicitely) by specifying a -basis, which is refered to as the *basis* of . If is the basis of , then the matrix with

is called the *basis matrix* of . We call the *generalized index* of . In case , the determinant is in fact equal to and is called the *index* of . The matrix is called the *Minkowski matrix* of .