Mathematical Sciences: Hopf Algebras and Galois Module Theory

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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.