TY - JOUR

T1 - Infinite hilbert class field towers over cyclotomic fields

AU - Shparlinski, Igor E.

N1 - Copyright 2008 Cambridge University Press. Article originally published in Glasgow Mathematical Journal, Volume 50, Issue 1, pp. 27-32. The original article can be found at http://dx.doi.org/10.1017/S0017089507003977

PY - 2008/1

Y1 - 2008/1

N2 - We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field Q2 pi im))has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over Q2 pi im))for some pm0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of Q2 pi i/m).

AB - We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field Q2 pi im))has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over Q2 pi im))for some pm0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of Q2 pi i/m).

UR - http://www.scopus.com/inward/record.url?scp=38149006399&partnerID=8YFLogxK

U2 - 10.1017/S0017089507003977

DO - 10.1017/S0017089507003977

M3 - Article

AN - SCOPUS:38149006399

VL - 50

SP - 27

EP - 32

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 0017-0895

IS - 1

ER -