The smallest prime that does not split completely in a number field

Li, Xiannan

doi:10.2140/ant.2012.6.1061

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The smallest prime that does not split completely in a number field

Xiannan Li

Vol. 6 (2012), No. 6, 1061–1096

DOI: 10.2140/ant.2012.6.1061

Abstract

We study the problem of bounding the least prime that does not split completely in a number field. This is a generalization of the classic problem of bounding the least quadratic nonresidue. Here, we present two distinct approaches to this problem. The first is by studying the behavior of the Dedekind zeta function of the number field near $1$ , and the second by relating the problem to questions involving multiplicative functions. We derive the best known bounds for this problem for all number fields with degree greater than $2$ . We also derive the best known upper bound for the residue of the Dedekind zeta function in the case where the degree is small compared to the discriminant.

Keywords

primes, split, number fields, Dedekind zeta function

Mathematical Subject Classification 2000

Primary: 11N60

Secondary: 11R42

Milestones

Received: 22 June 2010

Revised: 8 September 2011

Accepted: 26 September 2011

Published: 12 August 2012

Authors

Xiannan Li
Department of Mathematics Stanford University Stanford 94305 United States

Vol. 6, No. 6, 2012