Division Algebras and Field Invariants

NSF提案：“除法代数和域不变量”P.I.：大卫J.索特曼整数的分数，我们都在早期的年级学习，形成一个系统称为“场”。但是这个域不够大，因为在这个所谓的“有理数”域中，你不能取2的平方根，也不能写下π。因此，我们需要研究所有领域，这已成为一个不言而喻的真理。但是，即使是对所有场的研究也不够全面，因为在物理学中，或者在使用矩阵时，人们发现必须研究类场的对象，但它们是非交换的，这意味着A乘以B可能不等于B乘以A。这样的对象称为可除代数。这个问题，在最大意义上说，这格兰特认为是分类的所有divisionalgebras方面的更好地理解领域，特别是参考的“中心”的司代数，这是insidethe司代数，是一个领域。与中心相耦合的是一个除代数的“度”，它是一个正整数，用来度量除代数与中心相比有多大。除代数是一个古老的课题，当中心是一维时，已有大量的信息。例如，当中心靠近有理域时，任何除代数都被称为“循环的”，这意味着有一个很好的域描述。本文的重点是研究中心维数为2的可除代数.例如，major focus是中心来自ap-adic场上的曲线的情况，这是二维场的一个非常特殊的情况。在这种情况下，并非所有的除代数都是循环的，但提出者希望利用代数几何来证明，当除代数的次数为素数，且中心来自p-adic曲线时，循环性确实成立。也许研究除代数最重要的工具是伽罗瓦上同调，因为所有具有固定中心的除代数F的集合构成一个同构于具有单位系数的第二Galois上同调群的群。由于第二上同调可能很难计算，人们经常使用分歧来攻击这个群。更详细地说，任何离散赋值定义了一个到第一上同调群的映射，通过使用所有可能的离散赋值，人们希望捕获除代数。在特殊情况下，这是已知的工作，例如为合理的领域或领域所产生的p-adic曲线。在这两种情况下，一个显示，或希望显示，通过显示，有一个非循环伽罗瓦扩张，分裂所有的分歧，该司代数的周期性。这就引出了一个更根本的问题。如果D是一个素数次的除代数，是否总有一个相同次的循环域扩张，它至少能分裂所有的化？

NSF proposal: ``Division algebras and field invariants''P.I.: David J. SaltmanFractions of integers, which we all learned about in the early grades,form a system called a ``field''. But this field is not large enough,since in this so called ``rational'' field you cannot take the squareroot of 2, or write down pi. It has become, therefore, a truism that weneed to study all fields. But even the study of all fields is notencompassing enough, because in physics, or when using matrices, onefinds one must study objects which are field-like, but which arenoncommutative, meaning A times B might not equal B times A. Suchobjects are called division algebras. The question, in the largestsense, that this grant considers is the classification of all divisionalgebras in terms of the better understood fields, with particularreference to the ``center'' of the division algebra, which is insidethe division algebra and is a field. Coupled with the center is the``degree'' of a division algebra, which is a positive integer thatmeasures how much bigger the division algebras is as compared to itscenter. Division algebras are an old subject, with a great deal ofinformation known when the center is one dimensional. For example whenthe center is close to the rational field, any division algebra isknown to be so called ``cyclic'', which means there is a very gooddescription in terms of fields. The focus of this proposal is the studyof division algebras whose centers have dimension 2. For example, amajor focus is the case where the center comes from a curve over ap-adic field, a very special case of a 2 dimensional field. In thiscase, not all division algebras are cyclic, but the proposers hopes touse algebraic geometry to show that when the division algebra hasdegree a prime integer, and the center comes from a p-adic curve, thencyclicity does hold.Perhaps the most important tool in studying division algebras is Galoiscohomology, because the set of all division algebras with fixed centerF form a group isomorphic to the second Galois cohomology group withunit coefficients. Since second cohomology can be hard to compute,often one attacks this group by using ramification. In more detail, anydiscrete valuation defines a map to a first cohomology group, and byusing all possible discrete valuations one hopes to capture thedivision algebras. In special cases this is known to work, for examplefor rational field or the fields arising from p-adic curves. In bothcases, one shows, or hopes to show, cyclicity by showing there is acyclic Galois extension which splits all the ramification of thedivision algebra. This leads to a more fundamental question. If D is adivision algebra of prime degree, is there always a cyclic fieldextension, of the same degree, which at least splits all theramification?