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

Farshid Hajir, Christian Maire, Ravi Ramakrishna

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 $\rm F$. 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 $\# Γ$.

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

Abstract

We give a streamlined and effective proof of Ozaki's theorem that any finite -group is the Galois group of the -Hilbert class field tower of some number field . 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 with class number prime to . We construct by a sequence of -extensions ramified only at finite tame primes and also give explicit bounds on and the number of ramified primes of in terms of .
Paper Structure (9 sections, 4 theorems, 34 equations)

This paper contains 9 sections, 4 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Tools for the proof
  3. ${\mathbb F}_p[{\rm G}]$-modules and Minkowski Units
  4. Extensions ramified at a tame set of primes
  5. Solving the ramified embedding problem with one tame prime
  6. Cohomological facts implying the persistence of Minkowski units
  7. Solving the embedding problem
  8. The Stability Theorem
  9. The inductive step

Key Result

Theorem 1

Let ${\Gamma}$ be a finite $p$-group and ${\rm k}_0$ a number field with $(\#{\mathrm{Cl}}_{{\rm k}_0},p)=1$. There exist infinitely many number fields ${{\rm F}}/{\rm k}_0$ such that ${\mathrm{Gal}}({\rm L}_p({\rm F})/{\rm F}) \simeq {\Gamma}$ and

Theorems & Definitions (18)

  • Theorem
  • Corollary
  • proof
  • proof
  • Definition 1
  • proof
  • proof
  • proof
  • Definition 2
  • proof
  • ...and 8 more