Mathematical Sciences: Composition of Sums of Squares; Quadratic Forms and Galois Groups

本课题涉及二次型理论中的两个问题。第一个问题是关于平方和的合成律的存在性。第二个问题考虑了特征不同于两个的域上的二次型的行为与该域上可能出现的伽罗华群的类型之间的联系。本课题与二次型的经典研究有关。这一领域的早期成就之一是高斯发现每个正整数都是四个平方的和。这一领域不仅在纯数理论中很重要，而且在几何学和相对论数学中也很重要。

This project is concerned with two problems in quadratic form theory. The first concerns the existence of composition laws for sums of squares. The second problem considers the connections between the behavior of quadratic forms over a field of characteristic different than two and the types of Galois groups which can arise over that field. This project is related to the classical study of quadratic forms. One of the early achievements in this field is the discovery by Gauss that every positive whole number is the sum of four squares. This area is important not only in pure number theory but in geometry and the mathematics of relativity.