On Ozaki’s theorem realizing prescribed p-groups as p-class tower groups

Hajir, Farshid; Maire, Christian; Ramakrishna, Ravi

doi:10.2140/ant.2024.18.771

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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 $ℤ ∕ p$ -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

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Vol. 18, No. 4, 2024