Vol. 6, No. 6, 2012

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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

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.

primes, split, number fields, Dedekind zeta function
Mathematical Subject Classification 2000
Primary: 11N60
Secondary: 11R42
Received: 22 June 2010
Revised: 8 September 2011
Accepted: 26 September 2011
Published: 12 August 2012
Xiannan Li
Department of Mathematics
Stanford University
Stanford 94305
United States