Mathematical Sciences: Field Theory and Picard Groups

该奖项支持场论和交换代数方面的研究。主要研究者将研究伽罗瓦域扩展的乘法结构与相关群和不变量之间的关系。他还将研究在不一定是代数封闭的域上计算仿射环的皮卡德群的方案。基域有限扩展的乘法结构对计算有很大的影响。环是一个代数对象，在它上面定义了加法和乘法。环上的每个元素a都有一个加性逆，通常记作-a。然而，元素不需要有乘法逆。其中每个非零元素都有一个乘法逆的环称为域。这个项目关注一个字段的乘法结构和它的子字段的乘法结构之间的关系。它涉及到场论、群论和数论技术的结合。

This award supports research in field theory and commutative algebra. The principal investigator will study the relationship between the multiplicative structure of field extensions which are Galois and certain associated groups and invariants. He will also work on a scheme for computing Picard groups of affine rings over fields that are not necessarily algebraically closed. The computations are heavily influenced by the multiplicative structure of the finite extensions of the base field. A ring is an algebraic object having both an addition and a multiplication defined on it. Every element a in the ring has an additive inverse usually denoted -a. However, elements are not required to have multiplicative inverses. A ring in which every nonzero element has a multiplicative inverse is called a field. This project concerns the relationship between the multiplicative structure of a field and that of its subfields. It involves a combination of techniques from field theory, group theory and number theory.