标题： 关于p类域的三阶段塔的判别准则 显示英文标题

标题： Criteria for three-stage towers of p-class fields

摘要： 设 \( p \) 是一个素数，\( K \) 是一个具有非平凡 \( p \)-类群 \( \text\{Cl\}(p,K) \) 的数域。识别 \( K \) 的最大无分歧 \( \mathfrak\{p\} \)-扩张的伽罗瓦群 \( G = G(p,K) \) 的关键步骤是确定其两阶段逼近 \( M = G(p,2,K) \)，即第二导商 \( M = G/G'' \)。 所有度数为 \( p \) 的 \( K \) 的非分歧循环扩张 \( L/K \) 的 \( p \)-类群 \( \text\{Cl\}(p,L) \) 的阿贝尔型不变量族 \( \tau(1,K) \) 被称为 \( K \) 的指数-\( p \) 阿贝尔化数据（IPAD）。它能够确定 \( K \) 的第二 \( p \)-类群 \( M \) 的有限组竞争者。本文引入了两种不同类型的广义 IPAD 来获得更精细的结果。 多层 IPAD \( (\tau(1,K),\tau(2,K)) \) 包含度数为 \( p^2 \) 的非分歧阿贝尔扩张 \( L/K \) 的数据，并在 \( \text\{Cl\}(p,K) = (p,p,p) \) 的情况下允许更精确的 \( M \) 的阶数界，由于内存限制，当前 \( p \)-群生成算法的实现无法为此 \( M \) 产生明确的竞争者。 二阶迭代 IPAD \( \tau^\{(2)\}(K) \) 包含关于度数为 \( p^2 \) 或甚至 \( p^3 \) 的非阿贝尔非分歧扩张 \( L/K \) 的信息，并且可以识别 \( K = Q(\sqrt\{d\}) \) 的各种无限序列的二次域的 \( p \)-类塔群 \( G \)，当 \( \text\{Cl\}(p,K) = (p,p) \) 并且 \( p \)-类场塔的精确长度 \( L(p,K) = 3 \) 时，这是一项令人瞩目的新颖成果。 显示英文摘要

摘要： Let p be a prime and K be a number field with non-trivial p-class group Cl(p,K). A crucial step in identifying the Galois group G=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K), that is the second derived quotient M=G/G". The family tau(1,K) of abelian type invariants of the p-class groups Cl(p,L) of all unramified cyclic extensions L/K of degree p is called the index-p abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD ((tau(1,K),tau(2,K)) includes data on unramified abelian extensions L/K of degree p^2 and enables sharper bounds for the order of M in the case Cl(p,K)=(p,p,p), where current implementations of the p-group generation algorithm fail to produce explicit contestants for M, due to memory limitations. The iterated IPAD of second order tau^(2)(K) contains information on non-abelian unramified extensions L/K of degree p^2, or even p^3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(squareroot(d)) with Cl(p,K)=(p,p) possessing a p-class field tower of exact length L(p,K)=3 as a striking novelty.