Division Algebras and Invariant Fields

SALTMAN教授提议研究有关除法代数和不变域的问题。在除法代数领域，他将研究所有涉及描述这些对象的方法的问题。一系列问题涉及具有任意场的特殊的，所谓的半直积除法代数。他在问它们是不是更特殊的循环代数。另一系列的问题涉及考虑具有特殊基底域的一般除法代数，在这种情况下是p-adics上曲线的函数域，并询问它们是否循环。Saltman教授研究的第二个领域，不变场，也将分为两部分。第一个系列的问题是试图用伽罗瓦上同调来证明这些场中的某些是非理性的。第二个系列的问题涉及简单地研究这些不变域，例如，研究以这些不变域为中心的除法代数。除法代数是简单的对象，已经被研究了一百多年。基本思想是考虑有限维向量空间和一个表现良好的乘积。也就是说，这个乘积是结合式的，并且在实数中，每个非零元素都有一个逆，但是这个乘积是非交换的（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，这些中心是我们更熟悉的“领域”，比如理性或实数。描述所有有固定中心的除法代数实际上是在说一些关于这个中心的算术问题。有“一般”或非常一般的除法代数，它们的性质在某些方面反映了所有除法代数的性质。然而，它们的中心是难以理解的所谓不变场。人们可以直接研究这些不变域，从而更多地了解除法代数。在这项研究中，我们可以使用一系列数学工具，包括代数几何、伽罗瓦上同调和代数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.