Division Algebras and Invariant Fields

Saltman 9970213 Saltman教授建议研究有关除法代数和不变域的问题。在该地区的司代数，他将研究的问题，所有涉及的方式来描述这些对象。一系列的问题涉及特殊的，所谓的半直积，除代数与任意基域。他在问它们是否是更特殊的循环代数。另一系列的问题涉及考虑一般司代数与特殊的地面领域，在这种情况下功能领域的曲线p-adics，并询问他们是否循环。Saltman教授的第二个研究领域，不变场，也将有两条链。第一系列的问题涉及试图使用伽罗瓦上同调，以显示某些领域是非理性的。第二系列的问题涉及简单地研究这些不变领域，例如，调查部门代数与这些不变领域为中心。除代数是一个简单的对象，已经研究了一百多年。其基本思想是考虑有限维向量空间的产品是良好的行为。也就是说，乘积是结合的，并且和真实的数一样，每个非零元素都有逆元素，但是乘积是非交换的（即a乘以B不是B乘以a）。事实证明，这些对象体现了它们中心的深层属性，这些属性是更熟悉的“场”，如有理数或实数。描述所有具有固定中心的除代数实际上是在说一些关于这个中心的算术的深层次的东西。有“一般的”或非常一般的除代数，其性质在某些方面反映了所有除代数的性质。然而，它们的中心很难理解所谓的不变场。人们可以直接研究这些不变域，从而对除代数有更多的了解。人们可以使用，作为工具，在这项研究中，整个阵列的数学，包括代数几何，伽罗瓦上同调，代数K理论。

SALTMAN 9970213Professor Saltman proposes to study questions concerning division algebras and invariant fields. In the area of division algebras, he will study questions that all involve ways of describing these objects. One series of questions concern special, so called semidirect product, division algebras with arbitrary ground field. He is asking whether they are even more special cyclic algebras. Another series of questions in involve considering general division algebras with special ground fields, in this case function fields of curves over p-adics, and asking whether they are cyclic. The second area of study of Professor Saltman, invariant fields, will also have two strands. The first series of questions concern trying to use Galois cohomology to show certain of these fields are nonrational. The second series of questions involve simply studying these invariant fields and, for example, investigating division algebras with these invariant fields as center. Division algebras are simple objects that have been studied for over a hundred years. The basic idea is to consider finite dimensional vector spaces with a product that is well behaved. That is, the product is associative and, as in the real numbers, every nonzero element has an inverse, but the product is noncommutative (i.e. a times b is not b times a). It turns out the these objects embody deep properties of their centers which are the more familiar ``fields'' like the rationals or the reals. Describing all division algebras with a fixed center is really saying something deep about the arithmetic of this center. There are ``generic'' or very general division algebras whose properties in some ways reflect the properties of all division algebras. Their centers are, however, hard to understand so called invariant fields. One can directly study these invariant fields and thereby understand more about division algebras. One can use, as tools in this study, a whole array of mathematics including algebraic geometry, Galois cohomology, and algebraic K-theory.