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 $|G|$. In this article, we reduce that degree by taking the Frattini structure