Towers of p-class fields over algebraic number fields

The main research line of this project investigates unramified extensions of prime power degree over algebraic number fields. Building on our preparatory work on identifying the second p-class group of a number field by means of the structure of kernels and targets of transfers from this group to its normal subgroups containing the commutator subgroup we strive for proofs of restrictive selection rules for the second p-class group of special kinds of base fields. In international cooperation we plan to analyze bicyclic biquadratic and cyclic quartic number fields with elementary abelian p-class group of rank two or three, for the first time. The group theoretic information will arise from the arithmetic of these base fields, namely from the structure of the p-class groups of their unramified extensions within the first Hilbert p-class field and from the capitulation of ideal classes in these extensions. Even more challenging results we intend in the further investigation of the quadratic base fields, whose second p-class groups are already known from our preparatory work and thus provide an optimal starting point to tackle the last conceivable target, namely the length of the p-class field tower and the maximal unramified pro-p extension, by means of the methods of Golod-Shafarevich and Koch-Venkov. For the final deduction of selection rules for the second p-class group and for the potentially infinite pro-p group of the entire tower we stay in close contact with the international research frontier of the coclass theory of p-groups and determine the possible Galois groups with the aid of a combination of the p-group generation algorithm in bottom-up direction and the cohomological periodicity isomorphisms by forming quotients of infinite pro-p groups in top-down direction. In a second research line, we want to employ extensible results of our former Erwin Schrödinger project to the further investigation of ramified cyclic extensions of odd prime degree over cyclotomic and quadratic base fields, on the one hand, to obtain a classification of pure quintic fields according to their ambiguous principal ideals, on the other hand, for a deeper treatment of dihedral fields, using the integral representations of their S-unit groups as Galois modules, for finite Galois-stable sets S of places containing the Archimedean ones.

The class field tower consists of successive extensions of a base field. Every stage is associated with a group structure. The principal target of this re- search project was the identification of the tower group, which corresponds to the complete tower. International cooperation enabled the final clarification of the unsolved problem for a base field with class rank two. It turned out that these towers are always of finite height, whereas they reveal unbounded growth for rank three and more. A remarkable difference appeared for complex base fields with veritable skyscrapers as towers in contrast to real base fields which are contented with rather modest heights already. The project has initiated a new era of investigating three-stage towers thereby advancing Austrian science to the forefront of international research. Fruitful impact is to be expected on the neighbor area of group theory, since the determination of the tower group is based on a new kind of periodically repeating patterns in tree diagrams. A further fundamental insight is the tendency that most components of systems of type invariants become stable with fixed values, whereas some distinguished polarized constituents remain variable and encapsulate valuable key information on the tower group. The achievements of the project have been presented at various international scientific conferences in Scotland, California, Morocco, Hungary, Slovakia and Shanghai. Joint work with mathematicians in the United States, Morocco, Japan and Australia has consolidated the international cooperation with Austria.