Lower bounds for the least prime in Chebotarev

Fiori, Andrew

doi:10.2140/ant.2019.13.2199

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Lower bounds for the least prime in Chebotarev

Andrew Fiori

Vol. 13 (2019), No. 9, 2199–2203

DOI: 10.2140/ant.2019.13.2199

Abstract

In this paper we show there exists an infinite family of number fields $L$ , Galois over $ℚ$ , for which the smallest prime $p$ of $ℚ$ which splits completely in $L$ has size at least ${(log (| D_{L} |))}^{2 + o (1)}$ . This gives a converse to various upper bounds, which shows that they are best possible.

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Keywords

Chebotarev, class groups

Mathematical Subject Classification 2010

Primary: 11R44

Secondary: 11R29

Milestones

Received: 1 March 2019

Revised: 8 May 2019

Accepted: 27 June 2019

Published: 7 December 2019

Authors

Andrew Fiori
Department of Mathematics and Computer Science University of Lethbridge AB Canada

Vol. 13, No. 9, 2019