RUI: Galois Module Structure of Galois Cohomology

这个项目利用Lyndon-Hochschild-Serre精确序列来寻求Bloch-Kato猜想的场论结果。前人的结果已被用于推广Galois上同调中的Schreier公式，并建立了与绝对Galois群、Demuskin群和初等型猜想的prop商的上同调维的联系。本科生部分，包括本科生继续参与研究和向本科生提供关于伽罗瓦理论教学的MAA微型课程，将加强国家课程基础设施。合著的一本书，提供了从Kummer理论到Galois模结构的容易的过渡，将向更广泛的读者介绍这个主题，首先讨论模为素数的域的乘法群的结构，然后更广泛地讨论Galois上同调和Milnor K-理论。代数和数论的学科可以追溯到希腊人，并且在理解今天使用的被称为域的数系方面仍然提供了巨大的挑战。尽管场的多样性和复杂性令人望而生畏，但可以通过研究场的重新排列来研究场的结构：在不改变某些计算的结论的情况下，场中的一个数字可能被另一个数字替换的方式。一个令人震惊的事实是，从对这些可能的重新安排的了解中，可能会了解到大量关于油田本身的信息。这一观点可以追溯到不到两个世纪前，导致今天寻找每个领域在逻辑上可能的所有重新安排。对于代数学科来说，这一探索的完成很像将生物世界划分为属和种，确定元素周期表，甚至理解所有现有的基因。二十多年前，布洛赫和加藤在上个世纪曾简单地指出-但相当神秘-场结构的逻辑限制是由布洛赫和加藤猜测的。最近的研究证实，这些限制实际上是成立的，这使它们有点像是油田内部操作的规则手册。因此，对于场论来说，这是一个令人兴奋的时代，最终的结果不仅包括解决数学本身的其他问题，还包括相关学科的新视角，如密码学。

DMS 0600122John SwallowThis 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.