Explicit class field theory for rational function fields

[摘要] Class field theory describes the abelian extensions of a given field K in terms of variousclass groups of K, and can be viewed as one of the great successes of 20th centurynumber theory. However, the main results in class field theory are pure existenceresults, and do not give explicit constructions of these abelian extensions. Suchexplicit constructions are possible for a variety of special cases, such as for the field Qof rational numbers, or for quadratic imaginary fields. When K is a global functionfield, however, there is a completely explicit description of the abelian extensions ofK, utilising the theory of sign-normalised Drinfeld modules of rank one. In this thesiswe give detailed survey of explicit class field theory for rational function fields overfinite fields, and of the fundamental results needed to master this topic.

[发布机构] Stellenbosch University