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Tamely Ramified Towers and Discriminant Bounds for Number Fields

Published online by Cambridge University Press:  04 December 2007

Farshid Hajir  and
Christian Maire
Farshid Hajir
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, U.S.A. E-mail: fhajir@math.ucla.edu
Christian Maire
Affiliation:
Laboratoire A2X, Université Bordeaux I, Cours de la Libération, 33405 Talence Cedex, France. E-mail: maire@math.u-bordeaux.fr
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Abstract

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The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infm R2m. One knows that α0 [ges ] 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 [ges ] 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

Keywords

Root discriminantdiscriminant boundsclass field towersGolod–Shafarevich
Type
Research Article
Information
Compositio Mathematica , Volume 128 , Issue 1 , August 2001 , pp. 35 - 53
Copyright
© 2001 Kluwer Academic Publishers