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

TL;DR

This paper provides a simplified and effective proof of Ozaki's theorem, showing that any finite p-group can be realized as the Galois group of a p-class tower over some number field, extending the original result to broader settings.

Contribution

The authors present a streamlined proof of Ozaki's theorem applicable in mixed signature cases and construct explicit examples with bounds on degrees and ramification.

Findings

Any finite p-group can be realized as a Galois group of a p-class tower.

Constructs explicit number fields with controlled ramification and degree bounds.

Extends Ozaki's theorem beyond totally complex fields to broader contexts.

Abstract

We give a streamlined and effective proof of Ozaki's theorem that any finite $p$-group $\Gamma$ is the Galois group of the $p$-Hilbert class field tower of some number field $\rm F$. 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 ${\rm k}_0$ with class number prime to $p$. We construct ${\rm F}/{\rm k}_0$ by a sequence of ${\mathbb Z}/p$-extensions ramified only at finite tame primes and also give explicit bounds on $[{\rm F}:{\rm k}_0]$ and the number of ramified primes of ${\rm F}/{\rm k}_0$ in terms of $\# \Gamma$.