PreprintarXiv licencearXiv
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 pn, we prove τp(G) ≤ pℓΦ(G)+ ⌈logp(binomn+22+1)⌉. Here τp(G) is the least degree of a number field realizing G as its p-class tower group, and ℓΦ(G) is the iterated Frattini length of G. Thus, for groups of bounded Frattini length, the order-scale bound pn is replaced by the quadratic bound Op(n2). For Em=(Z/pZ)m, we further prove the sharp estimate τp(Em)asympp m2.
Related
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- Counting Nilpotent Galois extensions with prescribed genus number
- On the maximal index of C4--free unbalanced signed graphs of prescribed size
- Quantitative chain extensions and integral cohomological fillings
- Quantitative finiteness for maximal complex totally geodesic submanifolds of non-arithmetic ball quotients
BibTeX
@misc{kim2026maximal,
title = {{Maximal Unramified $p$-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem}},
author = {Kwang-Seob Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22766},
eprint = {2609.22766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22766},
}
RIS
TY - UNPB
TI - Maximal Unramified p-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem
AU - Kwang-Seob Kim
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22766
UR - https://arxiv.org/abs/2609.22766
AB - 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) ≤ p^ℓ_Φ(G)+ ⌈log_p(binomn+22+1)⌉. Here τ_p(G) is the least degree of a number field realizing G as its p-class tower group, and ℓ_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/pZ)^m, we further prove the sharp estimate τ_p(E_m)asymp_p m^2.
ER -