Winter School on Galois Theory, Volume 1

Galois Theory plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry. This special volume of the Luxembourg based peer-reviewed mathematics journal "Travaux mathématiques" unites two instructional texts that have grown out of lectures delivered at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. The contribution by Wulf-Dieter Geyer is about "Field Theory". It can be considered as a textbook in its own right. It manages to start at the level that any student possesses after any introductory algebra course and nevertheless to lead the reader to very advanced field theory at the frontier of current research, and to cover a wealth of material. Many examples are contained, which nicely enlighten the presented concepts, very often providing counterexamples that show why certain hypotheses are necessary. One also finds a chapter on the history of field theory as well as other historical remarks throughout the text. The second contribution addresses "Profinite Groups". It is written by Luis Ribes, who is the author of two standard books on this subject. Being necessarily much shorter than the two books, it has the feature of presenting an overview stressing the main concepts and the links with Galois Theory. Since for those proofs which are not included precise references are given, the notes, due to their conciseness and nevertheless great amount of material, constitute an excellent starting point for any Master or PhD student willing to learn this subject.