Theoretical and Experimental Approach to p-Class Field Towers of Cyclic Cubic Number Fields

Abstract

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.