RUI: Galois Module Structure of Galois Cohomology

DMS 0600122 John SwallowThis project engages the PI and collaborators at all levels in question to derive field-theoretical consequences of the Bloch-Kato Conjecture using the Lyndon-Hochschild-Serre exact sequence. 以前的结果已经被应用于推广Schreier公式在伽罗瓦上同调和建立联系的pro-pentations的上同调维数的绝对伽罗瓦群，Demuskin群，和与基本类型猜想。本科部分，包括继续参与研究的本科生和提供MAA微型课程的伽罗瓦理论教学本科生，将加强国家课程基础设施。合著的文本，提供了一个从库默理论到伽罗瓦模结构的可访问的过渡，将向更广泛的受众介绍这个主题，首先处理域模素数的乘法群的结构，然后更一般地处理伽罗瓦上同调和Milnor K-理论。代数和数论的学科可以追溯到希腊人，仍然在理解数字系统方面提供了实质性的挑战，也就是今天使用的领域 尽管场的多样性和复杂性令人生畏，但可以通过研究它们的重排来研究它们的结构：研究一个场中的一个数可以用另一个数代替的可能方式，而不改变某些计算的结论。一个令人吃惊的事实是，仅仅从这些可能的重新排列的知识中，就可以学到关于场本身的许多东西。这种观点可以追溯到不到两个世纪以前，它导致了今天对每个场的所有逻辑上可能的重新排列的探索。对于代数学科来说，完成这一探索就像将生物世界分类为属和种，确定元素周期表，甚至理解所有现有的基因。二十多年前，有些人简单地陈述了-但仍然相当神秘-场结构的逻辑限制在上个世纪被布洛赫和加藤阐明。最近的研究表明，这些限制确实存在，使其成为油田内部操作的管理手册。因此，这对场论来说是一个激动人心的时刻，最终的结果不仅包括解决数学中的其他问题，还包括相关学科的新观点，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.