Download this article
 Download this article For screen
For printing
Recent Issues

Volume 18
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
On Ozaki's theorem realizing prescribed $p$-groups as $p$-class tower groups

Farshid Hajir, Christian Maire and Ravi Ramakrishna

Vol. 18 (2024), No. 4, 771–786
DOI: 10.2140/ant.2024.18.771
Abstract

We give a streamlined and effective proof of Ozaki’s theorem that any finite p-group Γ is the Galois group of the p-Hilbert class field tower of some number field F . Our work is inspired by Ozaki’s and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field k 0 with class number prime to p. We construct F k 0 by a sequence of p-extensions ramified only at finite tame primes and also give explicit bounds on [F : k 0] and the number of ramified primes of F k 0 in terms of #Γ.

Keywords
Class group, Hilbert class field tower, Minkowski unit, $p$-groups
Mathematical Subject Classification
Primary: 11R29
Milestones
Received: 18 April 2022
Revised: 3 April 2023
Accepted: 29 May 2023
Published: 26 February 2024
Authors
Farshid Hajir
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States
Christian Maire
FEMTO-ST Institute
Universite Bourgogne Franche-Comté, CNRS
Besançon
France
Ravi Ramakrishna
Department of Mathematics
Cornell University
Ithaca, NY
United States

Open Access made possible by participating institutions via Subscribe to Open.