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Hilbertian fields and Galois representations ; ; Wiese, Gabor in Journal für die Reine und Angewandte Mathematik (2016), 712 We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising ... [more ▼] We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties. [less ▲] Detailed reference viewed: 120 (9 UL)Winter School on Galois Theory, Volume 2 Wiese, Gabor ; Arias De Reyna Dominguez, Sara ; Book published by University of Luxembourg / Campus Kirchberg (2013) 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 ... [more ▼] 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 four 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. It also includes one research article. Gebhard Böckle's contribution is a quite comprehensive survey on Galois representations. It focusses on the key ideas, and the long list of recommended references enables the reader to pursue himself/herself any of the mentioned topics in greater depth. Michael Schein's notes sketch the proof due to Khare and Wintenberger of one of the major theorems in arithmetic algebraic geometry in recent years, namely Serre's Modularity Conjecture. Moshe Jarden's contribution is based on his book on algebraic patching. It develops the method of algebraic patching from scratch and gives applications in contemporary Galois theory. David Harbater's text is complementary to Jarden's notes, and describes recent applications of patching in other aspects of algebra, for example: differential algebra, local-global principles, quadratic forms, and more. The focus is on the big picture and on providing the reader with intuition. The research article by Wulf-Dieter Geyer and Moshe Jarden concerns model completeness of valued PAC fields. [less ▲] Detailed reference viewed: 136 (9 UL)Winter School on Galois Theory, Volume 1 Wiese, Gabor ; Arias De Reyna Dominguez, Sara ; Book published by University of Luxembourg / Campus Kirchberg (2013) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 168 (20 UL) |
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