  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 $\# \Gamma$.

Number Theory