Research in Galois Theory and Integrating Technology in Undergraduate Mathematics

这个职业奖支持约翰·斯沃洛教授的研究和专业发展。这个项目有两个主要目标：找到新的建设性的方法来实现组作为伽罗瓦组，并推进技术在本科数学课堂上的使用。对于第一个目标，研究将集中在中央嵌入问题的障碍。该项目将提供新的见解伽罗瓦上同调和计算的作用，在展示显式多项式与给定的伽罗瓦群在有理数领域。教育计划的一个具体目标是编写一本关于伽罗瓦理论的本科教科书，该教科书将采用现代代数计算来有效地呈现示例，并将用最近在领域的显式构造领域的研究取代尺子和指南针构造作为基本主题。 该项目的研究部分是在数论的一般领域。数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

This CAREER award supports the research and professional development of Professor John Swallow. This project has two primary goals: to find new constructive methods to realize groups as Galois groups, and to advance the use of technology in the undergraduate mathematics classroom. For the first goal, the study will focus on obstructions to central embedding problems. The project will provide new insights into the role of Galois cohomology and computation in exhibiting explicit polynomials with given Galois groups over the field of rational numbers. One specific aim of the educational plan is to write an undergraduate textbook on Galois Theory which will employ modern algebraic computation to present examples efficiently and will replace ruler-and-compass constructions with recent research in the area of explicit construction of fields as an underlying theme. The research component of this project is in the general area of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.