A study on role of dedekind domains in algebraic number theory

Current study explains the role of Dedekind Domains in Algebraic Number Theory. Study was based on the literature and descriptive in nature. This paper introduces the important concept of Dedekind domains, which is important to understand the concept of rings of integers. Rings of integers arc an important class of Dedekind domains, but other examples include rings of polynomial functions on smooth algebraic curves. In this study, we prove two fundamental facts about Dedekind domains: every nonzero ideal can be factored uniquely as a product of prime ideals; and the set of ‘fractional ideals’ form a group under multiplication. We prove these statements by examining the local structure of Dedekind domains. The group structure allows us to introduce the "ideal class group ’ ’, which measures how far a Dedekind domain is from being a UFD, and plays a fundamental role throughout algebraic number theory.