Class Field Theory

Research Paper (postgraduate) from the year 2011 in the subject Mathematics - Number Theory grade: Postgraduate University of Sheffield language: English abstract: This is the first in a two part series of papers establishing (with proof) the main theorems of global class field theory. We first recap some of the main ideas of algebraic number theory using these to develop the Artin reciprocity law and the Takagi existence theorem both in terms of ideals and ideles. Finally we use the Hilbert class field in order to study the well known problem of which prime numbers can be represented in the form x^2 + ny^2 for integers xy and positive integer n.