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

Abstract

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 $|G|$. 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 \[ τ_p(G) \leq p^{\ell_Φ(G)+ \left\lceil\log_p\left(\binom{n+2}{2}+1\right)\right\rceil}. \] Here $τ_p(G)$ is the least degree of a number field realizing $G$ as its $p$-class tower group, and $\ell_Φ(G)$ 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 $E_m=(\Z/p\Z)^m$, we further prove the sharp estimate $τ_p(E_m)\asymp_p m^2$.