Division Algebras and Field Invariants

美国国家科学基金会的建议是：除法代数和场不变量。P.I.：大卫·J·萨尔特曼整数的分式，我们在早期的年级就知道了，它形成了一个叫做‘场’的系统。但是这个域不够大，因为在这个所谓的“有理”域中，你不能取2的平方根，也不能写下圆周率。因此，这已成为一个需要研究所有领域的老生常谈。但即使是对所有领域的研究也不够复杂，因为在物理学中，或者在使用矩阵时，人们发现必须研究类似于领域但不是可交换的对象，这意味着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。在最广泛的意义上，这项授权考虑的问题是根据更好地理解的域对所有除法代数进行分类，特别是指除法代数的“中心”，它在除法代数中，是一个域。与中心相连的是除法代数的“度”，这是一个正整数，用来衡量除法代数与其中心相比有多大。除法代数是一个古老的课题，当中心是一维的时候，有大量的信息是已知的。例如，当中心靠近有理域时，任何除法代数都被称为“循环”，这意味着对域有一个非常好的描述。这一建议的重点是研究中心为2维的除法代数。例如，主焦点是中心来自AP-ADIVE域上的曲线的情况，这是二维域的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，但作者希望利用代数几何来证明，当除法代数的阶数是一个素数，且中心来自一条p进曲线时，则循环性是成立的。也许研究除法代数最重要的工具是Galois上同调，因为所有以F为中心的除法代数的集合构成了一个同构于具有单位系数的第二个Galois上同调群的群。由于第二上同调可能很难计算，所以经常有人使用分支来攻击这个群。更详细地说，任何离散赋值定义了到第一上同调群的映射，并且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，这是已知可行的，例如，对于有理场或由p-进曲线产生的场。在这两种情况下，人们通过证明存在分裂除法代数的所有分支的非循环伽罗瓦扩张来展示或希望展示循环性。这引出了一个更根本的问题。如果D是素数阶除代数，是否总有一个相同次数的循环域扩张，它至少分裂了所有的分支？

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?