Theoretical Concept of Algebraic Number Theory

Exploring Algebraic Invariants and Applications in Number Fields

Abstract

In this paper we present about theoretical concept of algebraicnumber theory. Algebraic number theory studies algebraic properties of the ringof algebraic integers in a number field. We describe various algebraicinvariants of number fields, as well as their applications. These applicationsrelate to prime ramification, the finiteness of the class number, cyclotomicextensions, and the unit theorem.