Mathematical Sciences: Field Theory and Picard Groups

该奖项支持场论和交换代数的研究。主要研究伽罗华域扩张的乘法结构与某些相关群和不变量之间的关系。他还将致力于计算域上仿射环的Picard群的方案，这些仿射环不一定是代数闭域。计算受基场有限扩张的乘法结构的影响很大。环是定义了加法和乘法的代数对象。环中的每个元素a都有一个加法逆，通常表示为-a。但是，元素不一定要有乘法反转。其中每个非零元素都有一个乘法逆的环称为域。这个项目涉及一个域的乘法结构和它的子域的乘法结构之间的关系。它结合了场论、群论和数论的技巧。

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.