Maximal Unramified $p$-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem

Ozaki proved that every finite $p$-group occurs as the Galois group of a maximal unramified $p$-extension of a number field. Hajir, Maire and Ramakrishna made this theorem effective, obtaining a base-field degree of order $\mathrm{\mid }G\mathrm{\mid }$. In this article, we reduce that degree by taking the Frattini structure of $G$ into account. More precisely, for an odd prime $p$ and a finite $p$-group $G$ of order $p^n$, we prove $${\tau }_p(G)\le p^{{\mathrm{\ell }}_{\mathrm{\Phi }}(G)+\lceil {\log }_p(\left(\genfrac{}{}{0px}{}{n+2}{2}\right)+1)\rceil }\mathrm{.}$$ Here ${\tau }_p(G)$ is the least degree of a number field realizing $G$ as its $p$-class tower group, and ${\mathrm{\ell }}_{\mathrm{\Phi }}(G)$ is the iterated Frattini length of $G$. Thus, for groups of bounded Frattini length, the order-scale bound $p^n$ is replaced by the quadratic bound $O_p(n^2)$. For $E_m=(\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^m$, we further prove the sharp estimate ${\tau }_p(E_m){\asymp }_p{m}^2$.