Open Access

Distribution of orders in number fields

Research in the Mathematical Sciences20152:6

https://doi.org/10.1186/s40687-015-0027-8

Received: 14 August 2014

Accepted: 15 April 2015

Published: 10 June 2015

Abstract

In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techniques to analyze the zeta function. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field. We also obtain non-trivial bounds for higher degree number fields.

AMS Subject Classification: Primary 11M41; 11R29; secondary 11S40

Keywords

Orders p-adic and motivic integration Subring zeta function