Mathematical Sciences: Hopf Algebras and Galois Module Theory

设L为域K的有限扩展的局部或全局数域，当L/K为伽罗瓦时，经典伽罗瓦模理论试图将数域中的整数环理解为其伽罗瓦群环上的模。本研究将探讨这方面的几个问题。主要研究者将确定数域内整数环在其伽罗瓦群环上局部自由的适当准则。他也将研究可交换的，协交换的素数幂秩Hopf代数的分类。数论是研究整数性质的学科，是数学最古老的分支之一。在现代，数论中的问题为纯代数、分析和几何领域的新数学的创造提供了动力。数论的一些最新应用出现在理论计算机科学和编码理论中。

Let L be a local or global number field which is a finite extension of the field K. When L/K is Galois, classical Galois module theory seeks to understand the ring of integers in a number field as a module over its Galois group ring. This research will examine several questions in this vein. The principal investigator will determine suitable criteria for the ring of integers in a number field to be locally free over its Galois group ring. He will also work on the classification of commutative, cocommutative prime-power rank Hopf algebras. Number theory, which is the study of the properties of the whole numbers, is one of the oldest branches of mathematics. In modern days, problems in number theory have furnished the driving force for the creation of new mathematics in the fields of pure algebra, analysis, and geometry. Some of the most recent applications of number theory have appeared in theoretical computer science and coding theory.