Skip to content

Introduction

By definition, mathematically a number field is just a finite extension of the rational . In Hecke, a number field is recursively defined as being the field of rational numbers or a finite extension of a number field . In the second case, the extension can be defined in the one of the following two ways:

  • We have , where is an irreducible polynomial (simple extension), or
  • We have , where are univariate polynomials (non-simple extension).

In both cases we refer to as the base field of the number field . Another useful dichotomy comes from the type of the base field. We call an absolute number field, if the base field is equal to the rational numbers .