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。其中一系列问题涉及具有任意基域的特殊的、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。中的另一系列问题涉及考虑具有特殊基场的一般除法代数，在这种情况下，p-adics上的曲线的函数场，并询问它们是否是循环的。萨尔特曼教授的第二个研究领域--不变场--也将有两条线索。第一系列问题涉及试图使用伽罗华上同调来证明这些域中的某些域是无理的。第二系列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。除法代数是研究了一百多年的简单对象。其基本思想是考虑有限维向量空间的乘积表现良好。也就是说，乘积是结合的，并且像在实数中一样，每个非零元素都有一个逆，但乘积是非对易的(即a乘b不是b乘a)。事实证明，这些物体体现了它们中心的深层属性，这些中心是更熟悉的‘场’，比如有理数或实数。用一个固定的中心来描述所有除法代数，实际上是在深刻地说明这个中心的算术。存在“一般”或非常一般的除法代数，它们的性质在某种程度上反映了所有除法代数的性质。然而，它们的中心很难理解，即所谓的不变场。人们可以直接研究这些不变域，从而对除法代数有更多的了解。作为本研究的工具，我们可以使用包括代数几何、伽罗华上同调和代数K理论在内的一系列数学知识。

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.