RUI: Galois Module Structure of Galois Cohomology

Davidson College, Davidson NC

Investigators

Abstract

DMS 0600122 John Swallow This project engages the PI and collaborators at all levels in the quest to derive field-theoretic consequences of the Bloch-Kato Conjecture using the Lyndon-Hochschild-Serre exact sequence. Prior results have already been applied to generalize Schreier's formula in Galois cohomology and establish connections with the cohomological dimension of pro-p quotients of absolute Galois groups, with Demuskin groups, and with the Elementary Type Conjecture. Undergraduate components, including the continued involvement of undergraduates in research and the offering of MAA minicourses on the teaching of Galois theory to undergraduates, will enhance the national curricular infrastructure. A co-authored text, providing an accessible transition from Kummer theory to Galois module structure, will introduce the subject to a wider audience, treating the structure first of the multiplicative groups of fields modulo a prime and then more generally of Galois cohomology and Milnor K-theory. The disciplines of algebra and number theory reach back as far back as the Greeks and still offer substantial challenges in understanding the number systems, called fields, used today. Although the sheer variety and complexity of fields is daunting, their structure can be studied by investigating their rearrangements: the possible ways in which one number in a field may be substituted for another, without altering the conclusions of certain calculations. A startling fact is that from no more than the knowledge of these possible rearrangements, a great deal may be learned about the fields themselves. This point of view, going back less than two centuries, leads today to the search for all logically possible rearrangements of each field. For the discipline of algebra, the completion of this quest is much like the classification of the living world into genera and species, the determination of the periodic table of elements, or even the understanding of all existing genes. Over twenty years ago, some simply stated---but nevertheless fairly mysterious---logical limitations on the structure of fields were conjectured in the last century by Bloch and Kato. Very recent work has established that these limitations do in fact hold, making them something like a regulation manual for the internal operation of fields. As a result, this is an exciting time for field theory, and the eventual consequences will include not only the solution of other problems in mathematics proper but also new perspectives in related disciplines, such as cryptography.