Mathematical Sciences: Crossed Products over Rational Function Fields

本课题主要研究环论中的问题，并研究有限维除代数的极大子域问题。主要研究两个问题：(1)给定域K的有限代数扩张L，是否存在以L为极大子域的中心为K的有限维除代数？和(2)给定一个有限群G，它是K上的伽罗瓦群，是否存在K的伽罗瓦扩张L，且L/K的伽罗瓦群同构于G，且存在将GXG映射到L*的因子集f，使得交叉积代数(L/K，f)是除环？本项目所支持的研究属于环理论的一般领域。首席研究员将研究一个中心单代数如何包含另一个中心单代数。这项工作借鉴了群论中的一些深入结果，并将有助于揭示代数和数论中的各种问题。

This project is concerned with problems in ring theory and deals with questions about the maximal subfields of finite dimensional division algebras. The principal investigator will consider the following two questions: (1) given a finite algebraic extension L of a field K, does there exist a finite dimensional division algebra with center K containing L as a maximal subfield?, and (2) given a finite group G which occurs as a Galois group over K, does there exist a Galois extension L of K with the Galois group of L/K isomorphic to G and a factor set f mapping GXG into L* such that the crossed product algebra (L/K,f) is a division ring? The research supported in this project is in the general area of ring theory. The principal investigator will study how one central simple algebra can contain another. This work draws on some deep results in group theory and will shed light on various questions in algebra and number theory.