Title: Nilpotent Artin-Schreier theory Abstract: In this talk, we review techniques used for parametrizing extensions of fields of characteristic p, and we show how these techniques specialize to known theories (Artin-Schreier-Witt theory, ϕ-modules, ...). We then review the Lazard correspondence, which relates p-groups of nilpotency class smaller than p with Lie algebras. By combining these two…

Title: Higher ramification groups and counting Galois extensions Abstract: This is a follow-up to last week's talk on nilpotent Artin-Schreier theory. We recall some basic facts about higher ramification groups of Galois extensions and where they come up in class field theory. Then, we study how they fit in with Abrashkin's nilpotent Artin-Schreier theory, and we sketch how we can (for some…

Title: Classification of imaginary quadratic number fields with class number 1, part I Abstract: We give an overview of the problem of determining how many imaginary quadratic number fields have class number 1. We then present Heilbronn and Linfoot's result, that there are at most 10 such number fields.

Title: Classification of imaginary quadratic number fields of class number 1, part II Abstract: In this talk, we show how Baker's result on linear forms in logarithms of algebraic numbers provides a solution to the problem of classifying imaginary quadratic number fields of class number one, following the approach of Bundschuh and Hock.

Title: On Galois extensions with cyclic decomposition groups Abstract: Call a Galois extension of number fields "locally cyclic" if all its decomposition groups are cyclic. Such extensions are interesting by themselves, but are also of significance for other problems in and around inverse Galois theory. I will give a survey of several recent results by myself and others, in part on realizing…