Vol. 277, No. 1, 2015

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Upper bounds of root discriminant lower bounds

Siman Wong

Vol. 277 (2015), No. 1, 241–255
Abstract

For any rational number t [0,1], define the logarithmic Martinet function β(t) to be the liminf of the logarithm of the root discriminant of number fields K with r1(K)[K : ] = t as [K : ] goes to infinity. Under the generalized Riemann hypothesis for Dedekind zeta functions of number fields, we show that β(t) < 14.55 for a dense subset of rational numbers t [0,1]. We also study unconditional estimates of the growth of root discriminants by studying how the polynomial discriminant behaves under perturbation of coefficients, and by using Pisot numbers.

Keywords
Chebotarev density theorem, class field towers, Pisot numbers, root discriminants
Mathematical Subject Classification 2010
Primary: 11R29
Secondary: 11R37, 11R21
Milestones
Received: 23 January 2014
Revised: 13 February 2015
Accepted: 13 February 2015
Published: 6 August 2015
Authors
Siman Wong
Department of Mathematics and Statistics
University of Massachusetts
Lederle Graduate Research Tower
Box 34515
Amherst, MA 01003-9305
United States