On the structure of immediate extensions of valued fields

Abstract

The connection between a valued eld extension and the corresponding extensions of the value group and the residue eld is meaningful for the theory of valued elds. When this connection is interrupted, the structure of valued eld extensions is much more complicated. This causes one of the main hurdles to solve many important questions in valuation theory and related areas of mathematics. Crucial examples of this situation are defect extensions and immediate extensions of valued elds. A better understanding of both types of extensions turned out to be important for questions in algebraic geometry, like resolution of singularities, problems in real algebra and the model theory of valued elds. In this thesis we study the structure and constructions of immediate as well as defect extensions of valued elds. In particular, we focus on the structure of maximal immediate extensions of valued elds. In connection with local uniformization, a local version of resolution of singularities, we investigate the problems related to defect extensions. We describe properties of distances of elements in valued eld extensions, which turned out to be a useful tool for the study of the structure of defect extensions of valued elds of positive characteristic. We also give an upper bound of the number of distinct distances of immediate elements of a bounded degree. We further study the problem of existence of in nite towers of Galois defect extensions of prime degree. We give conditions for a valued eld to admit such towers and present constructions of them. In connection with questions related to local uniformization we present constructions of in nite towers of Artin-Scheier defect extensions of rational function elds in two variables over elds of positive characteristic. We consider the classi cation of Artin-Schreier defect extensions into \dependent" and \independent" ones (according to whether they are connected with purely inseparable defect extensions, or not). To understand the meaning of the classi cation for the issue of local uniformization, we consider various valuations of the above mentioned rational function elds and investigate for which they admit an in nite tower of Artin-Schreier defect extensions of each type. The existence of in nite towers of Galois defect extensions of prime degree turned out to be important for the structure of maximal immediate extensions of valued elds, which is the next problem treated in this thesis. We give conditions for a valued eld to admit maximal immediate extensions of in nite transcendence degree. This problem is tightly connected with the description of the possible extensions of a valuation from a given eld to an algebraic function eld. We further consider algebraic extensions of maximal elds and study the structure of immediate extensions of such elds. We also investigate the problem of uniqueness of maximal immediate extensions. We prove that there is a class of valued elds which admit an algebraic maximal immediate extension as well as one of in nite transcendence degree, which can be seen as the worst possible case of non-uniqueness. In our studies of maximal immediate extensions we consider also valued elds (K; v) with p-divisible value group and perfect residue eld, where p is the characteristic exponent of the residue eld Kv. Maximal immediate extensions of such elds are tame elds. Because of their good valuation theoretical and model theoretical properties, tame elds play an important role in the theory of valued elds and its applications. We discuss rst the case of elds with maximal immediate extensions of nite transcendence degree and describe the structure of such extensions. We then relate the existence of defect extensions of the eld (K; v) with the structure of the maximal immediate extensions of this eld. We prove that if the eld (K; v) admits a nontrivial separable-algebraic defect extension, then every maximal immediate extension of K is of in nite transcendence degree. We nally apply the results to the description of the structure of valued rational function elds. In particular, we give necessary and su cient conditions on (K; v) to admit an extension of the valuation to a rational function eld F over K such that vF=vK is a torsion group and the residue eld extension FvjKv is algebraic.