Classification of Prime Ideals in Integral Group Algebras of Finite Abelian Groups

Abstract

Let Z[G] be the integral group algebra of the group G. In this thesis, we consider the problem of determining all prime ideals of Z[G] where G is both finite and abelian. Because of Krull dimension arguments, there are only two types of prime ideals in Z[G]. First, we show that we can think of Z[G] as the quotient of a polynomial ring. Using this fact, and some Galois theory, we then classify the minimal prime ideals of our Z[G] where we restrict our group to having one or two generators. Next, we determine the form of the maximal ideals of Z[G] for the same case. However, the maximal ideals in our list need not be distinct. We further explore this issue restricting ourselves to cyclic groups. Using our previous work and cyclotomic field theory we are able to determine the duplication in our previous list.