Explicit counting of ideals in number fields of arbitrary degree

TL;DR

This paper develops explicit methods based on the geometry of numbers to count integral ideals in number fields, minimizing degree-related error terms by studying fundamental domains and applying lattice theory.

Contribution

It introduces improved explicit estimates for counting ideals in arbitrary degree number fields, reducing the impact of the degree on error terms.

Findings

Explicit bounds for the number of integral ideals in number fields.

Methods to choose fundamental domains that minimize error.

Adaptation of Schmidt's partition trick to general number fields.

Abstract

We implement methods from the geometry of numbers to give explicit estimates for the number of integral ideals in a number field. We pay particular attention to minimising the effect of the degree $n$ of the number field on the error term and avoid terms on the order of $n^{n^2}$. We do this by studying fundamental domains for the action of multiplying with units of infinite order in Minkowski space. With some lattice theory we show that one can make different choices for such a fundamental domain, which yield a smaller error, especially when the degree of the field extension is large. We also adapt Schmidt's partition trick to this generalised setting.