Algebra & Number Theory

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

Xiannan Li

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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.

Article information

Source
Algebra Number Theory, Volume 6, Number 6 (2012), 1061-1096.

Dates
Received: 22 June 2010
Revised: 8 September 2011
Accepted: 26 September 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729861

Digital Object Identifier
doi:10.2140/ant.2012.6.1061

Mathematical Reviews number (MathSciNet)
MR2968634

Zentralblatt MATH identifier
1321.11105

Subjects
Primary: 11N60: Distribution functions associated with additive and positive multiplicative functions
Secondary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

Keywords
primes split number fields Dedekind zeta function

Citation

Li, Xiannan. The smallest prime that does not split completely in a number field. Algebra Number Theory 6 (2012), no. 6, 1061--1096. doi:10.2140/ant.2012.6.1061. https://projecteuclid.org/euclid.ant/1513729861


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