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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

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.

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

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