论文标题
循环数字字段的P级现场塔的理论和实验方法
Theoretical and Experimental Approach to p-Class Field Towers of Cyclic Cubic Number Fields
论文作者
论文摘要
奇数质度的循环数字字段被构建为在理性数字字段上的射线类字段。它们被收集到共享共同指挥和判别的多重组中。该算法在岩浆中实现，并应用于所有环状和环状立方场，导体低于100000。我们的主要注意力专用于带有导体的两个或三个素数分隔线的环状立场理论。这些字段构成双重双线和四重奏。理论技术包括划分导体的数量之间的立方残留条件班级塔楼，通过Artin通过Artin的模式识别在有限群体的后代树上，有命令3，Shafarevich定理3级田间塔组的关系等级以及对塔组的Galois行动及其Metabelianization。严格的证据是针对三阶段塔的首次出现在循环立方场上，该塔具有基本的双环或三环或非元素双环的3级组。在实验上，除了少数复杂的八位字节外，第二个P级基团和P级塔的长度都是针对100000以下的所有导体和P = 2,3,5确定的。一个有趣的应用程序能够识别并实现Andozhskii和Tsvetkov的封闭组。
Cyclic number fields of odd prime degree are constructed as ray class fields over the rational number field. They are collected in multiplets sharing a common conductor and discriminant. The algorithms are implemented in Magma and applied to all cyclic quintic and cyclic cubic fields with conductors below 100000. Our primary attention is devoted to the theory of cyclic cubic fields with two or three prime divisors of the conductor. These fields form doublets and quartets. Theoretical techniques comprise cubic residue conditions between the primes dividing the conductor, the structure of 3-class groups of all components of doublets and quartets, Galois cohomology of unit groups and ambiguous principal ideals, absolute genus fields and their bicyclic bicubic subfields, class number relations, transfer kernels and abelian quotient invariants of unramified cyclic cubic extensions and their impact on the class field tower, pattern recognition via Artin transfers on descendant trees of finite groups with order a power of 3, the Shafarevich Theorem on the relation rank of the 3-class field tower group, and the Galois action on the tower group and on its metabelianization. Rigorous proofs are given for the first occurrences of three-stage towers over cyclic cubic fields with elementary bicyclic or tricyclic or non-elementary bicyclic 3-class group. Experimentally, the second p-class groups and the length of the p-class tower are determined for all conductors below 100000 and for p=2,3,5, with the exception of the few intricate octets. An interesting application is able to identify and realize the closed groups by Andozhskii and Tsvetkov.
