AI 中文总结
本文针对伽罗瓦群为非p群与循环p群直积的伽罗瓦扩张，建立了p处局部Minkowski单位的存在准则，应用该准则构造了满足条件的非阿贝尔扩张，分析了相关Iwasawa模结构并推广了研究范围。
AI 中文摘要
我们建立了p处局部Minkowski单位存在的判别准则，该准则适用于所有伽罗瓦群同构于非p群与循环p群直积的伽罗瓦扩张。作为应用，我们在不同分歧条件下构造了允许p处局部Minkowski单位的非阿贝尔扩张，分析了数域的Z_p扩张中单位的Iwasawa模结构，还将研究推广到p次单位根群非平凡的情形。
英文摘要
We establish a criterion for the existence of a local Minkowski unit at $p$ that applies to all Galois extensions with Galois group isomorphic to the direct product of a non-$p$-group and a cyclic $p$-group. As applications, we construct non-abelian extensions admitting a local Minkowski unit at $p$ under various ramification conditions and analyze the Iwasawa module structure of units in $\mathbb{Z}_p$-extensions of number fields. We also extend our study to the case where the group of $p$-power roots of unity is nontrivial.
Comments30 pages, Minor corrections and typo fixes
