Division Algebras and Field Invariants

NSF提案：“除法代数和场不变量”字体大卫·j·索尔特曼我们在小学的时候都学过整数的分数，它们构成了一个叫做“域”的系统。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，或者写出。因此，我们需要研究所有领域，这已成为一个不言自明的真理。但即使是对所有场的研究也不够全面，因为在物理学中，或者当使用矩阵时，人们发现必须研究类场的对象，但它们是不可交换的，也就是说A乘以B可能不等于B乘以A。这样的对象被称为除法代数。从最广泛的意义上说，该基金考虑的问题是根据更好理解的域对所有可除代数进行分类，特别涉及可除代数的“中心”，它位于可除代数内部，是一个域。与中心相结合的是除法代数的“度”，这是一个正整数，用于测量除法代数相对于其中心的大多少。除法代数是一门古老的学科，当中心是一维的时候，有很多已知的信息。例如，当中心靠近有理域时，任何除法代数都被称为“循环”，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的除法代数。例如，一个主要焦点是中心来自于一个曲线在一个二维场，一个非常特殊的情况。在这种情况下，并不是所有的除法代数都是循环的，但提议者希望使用代数几何来证明，当除法代数的次数是素数整数，并且中心来自p进曲线时，环性确实成立。也许在研究除法代数中最重要的工具是伽罗上同调，因为所有具有固定中心f的除法代数的集合形成了一个与具有单位系数的第二个伽罗上同调群同构的群。由于第二上同调很难计算，因此通常使用分支来攻击这个组。更详细地说，任何离散赋值都定义了到第一个上同群的映射，通过使用所有可能的离散赋值，人们希望捕获除法代数。在特殊情况下，这是已知的工作，例如有理域或由p进曲线产生的域。在这两种情况下，人们通过证明有一个非循环的伽罗瓦扩展来证明，或者希望证明，循环性，它可以把除法代数的所有分支分开。这就引出了一个更根本的问题。如果D是素数次的除法代数，是否总有一个相同次的循环场扩展，至少能将所有的化分开？

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?