具有给定 Galois 群的最大非分歧 $p$-扩张：Ozaki 定理的定量改进
Maximal Unramified $p$-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem
- Chosun University（朝鲜大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文改进了 Ozaki 定理，通过利用 Frattini 结构将实现有限 p-群为 p-类塔群所需的基域次数从阶尺度降至二次尺度，并给出初等交换群的精确估计。
AI中文摘要：
Ozaki 证明了每个有限 $p$-群都可以作为某个数域的最大非分歧 $p$-扩张的 Galois 群出现。Hajir、Maire 和 Ramakrishna 使该定理成为有效的，获得了阶为 $|G|$ 的基域次数。在本文中，我们通过考虑 $G$ 的 Frattini 结构来降低该次数。更精确地，对于奇素数 $p$ 和阶为 $p^n$ 的有限 $p$-群 $G$，我们证明 \\[ \tau_p(G) \leq p^{\ell_\Phi(G)+ \left\lceil\log_p\left(\binom{n+2}{2}+1\right)\right\rceil}. \\] 这里 $\tau_p(G)$ 是实现 $G$ 作为其 $p$-类塔群的最小数域次数，而 $\ell_\Phi(G)$ 是 $G$ 的迭代 Frattini 长度。因此，对于 Frattini 长度有界的群，阶尺度界 $p^n$ 被二次界 $O_p(n^2)$ 取代。对于 $E_m=(\Z/p\Z)^m$，我们进一步证明精确估计 $\tau_p(E_m)\asymp_p m^2$。
英文摘要：
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 $τ_p(E_m)\asymp_p m^2$.
