Introduction

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

  • We have L=K[x]/(f)L = K[x]/(f), where fK[x]f \in K[x] is an irreducible polynomial (simple extension), or
  • We have L=K[x1,,xn]/(f1(x1),,fn(xn))L = K[x_1,\dotsc,x_n]/(f_1(x_1),\dotsc,f_n(x_n)), where f1,,fnK[x]f_1,\dotsc,f_n \in K[x] are univariate polynomials (non-simple extension).

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

This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.