Let K be a quadratic imaginary number field, let Kp^(infinity) the top of its p-class field tower for p an odd prime, and let G=Gal(Kp^(infinity)/K). It is known, due to a tremendous collection of work ranging from the principal results of class field theory to the famous Golod-Shafarevich inequality, that G is finite if the p-rank of the class group of K is 0 or 1, and is infinite if this rank is at least 3. This leaves the rank 2 case as the only remaining unsolved case. In this case, while finiteness is still a mystery, much is still known about G: It is a 2-generated, 2-related pro-p-group equipped with an involution that acts as the inverse modulo commutators, and is of one of three possible Zassenhaus types (defined in the paper). If such a group is finite, we will call it an interesting p-tower group. We further the knowledge on such groups by showing that one particular Zassenhaus type can occur as an interesting p-tower group only if the group has order at least p^24 (Proposition 8.1), and by proving a succinct cohomological condition (Proposition 4.7) for a p-tower group to be infinite. More generally, we prove a Golod-Shafarevich equality (Theorem 5.2), refining the famous Golod-Shafarevich inequality, and obtaining as a corollary a strict strengthening of previous Golod-Shafarevich inequalities (Corollary 5.5). Of interest is that this equality applies not only to finite p-groups but also to p-adic analytic pro-p-groups, a class of groups of particular relevance due to their prominent appearance in the Fontaine-Mazur conjecture. This refined version admits as a consequence that the sizes of the first few modular dimension subgroups of an interesting p-tower group G are completely determined by p and its Zassenhaus type, and we compute these sizes. As another application, we prove a new formula (Corollary 5.3) for the Fp-dimensions of the successive quotients of dimension subgroups of free pro-p-groups.

Dissertation (PhD)--University of Stellenbosch, 2007. / Bibliography. Also available via the Internet.

Thesis (MSc)--Stellenbosch University, 2003 / ENGLISH ABSTRACT: A tower of global function fields :F = (FI, F2' ... ) is an infinite tower of separable extensions of algebraic function fields of one variable such that the constituent function fields have the same (finite) field of constants and the genus of these tend to infinity. A study can be made of the asymptotic behaviour of the ratio of the number of places of degree one over the genus of FJWq as i tends to infinity. A tower is called asymptotically good if this limit is a positive number. The well-known Drinfeld- Vladut bound provides a general upper bound for this limit. In practise, asymptotically good towers are rare. While the first examples were non-explicit, we focus on explicit towers of function fields, that is towers where equations recursively defining the extensions Fi+d F; are known. It is known that if the field of constants of the tower has square cardinality, it is possible to attain the Drinfeld- Vladut upper bound for this limit, even in the explicit case. If the field of constants does not have square cardinality, it is unknown how close the limit of the tower can come to this upper bound. In this thesis, we will develop the theory required to construct and analyse the asymptotic behaviour of explicit towers of function fields. Various towers will be exhibited, and general families of explicit formulae for which the splitting behaviour and growth of the genus can be computed in a tower will be discussed. When the necessary theory has been developed, we will focus on the case of towers over fields of non-square cardinality and the open problem of how good the asymptotic behaviour of the tower can be under these circumstances. / AFRIKAANSE OPSOMMING: 'n Toring van globale funksieliggame F = (FI, F2' ... ) is 'n oneindige toring van skeibare uitbreidings van algebraïese funksieliggame van een veranderlike sodat die samestellende funksieliggame dieselfde (eindige) konstante liggaam het en die genus streef na oneindig. 'n Studie kan gemaak word van die asimptotiese gedrag van die verhouding van die aantal plekke van graad een gedeel deur die genus van Fi/F q soos i streef na oneindig. 'n Toring word asimptoties goed genoem as hierdie limiet 'n positiewe getal is. Die bekende Drinfeld- Vladut grens verskaf 'n algemene bogrens vir hierdie limiet. In praktyk is asimptoties goeie torings skaars. Terwyl die eerste voorbeelde nie eksplisiet was nie, fokus ons op eksplisiete torings, dit is torings waar die vergelykings wat rekursief die uitbreidings Fi+d F; bepaal bekend is. Dit is bekend dat as die kardinaliteit van die konstante liggaam van die toring 'n volkome vierkant is, dit moontlik is om die Drinfeld- Vladut bogrens vir die limiet te behaal, selfs in die eksplisiete geval. As die konstante liggaam nie 'n kwadratiese kardinaliteit het nie, is dit onbekend hoe naby die limiet van die toring aan hierdie bogrens kan kom. In hierdie tesis salons die teorie ontwikkel wat benodig word om eksplisiete torings van funksieliggame te konstrueer, en hulle asimptotiese gedrag te analiseer. Verskeie torings sal aangebied word en algemene families van eksplisiete formules waarvoor die splitsingsgedrag en groei van die genus in 'n toring bereken kan word, sal bespreek word. Wanneer die nodige teorie ontwikkel is, salons fokus op die geval van torings oor liggame waarvan die kardinaliteit nie 'n volkome vierkant is nie, en op die oop probleem aangaande hoe goed die asimptotiese gedrag van 'n toring onder hierdie omstandighede kan wees.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / Explicit towers of algebraic function fields over finite fields are studied by considering their ramification behaviour and complete splitting. While the majority of towers in the literature are recursively defined by a single defining equation in variable separated form at each step, we consider towers which may have different defining equations at each step and with arbitrary defining polynomials. The ramification and completely splitting loci are analysed by directed graphs with irreducible polynomials as vertices. Algorithms are exhibited to construct these graphs in the case of n-step and -finite towers. These techniques are applied to find new tamely ramified n-step towers for 1 n 3. Various new tame towers are found, including a family of towers of cubic extensions for which numerical evidence suggests that it is asymptotically optimal over the finite field with p2 elements for each prime p 5. Families of wildly ramified Artin-Schreier towers over small finite fields which are candidates to be asymptotically good are also considered using our method.