Research Paper from the year 2011 in the subject Mathematics - Number Theory, grade: Postgraduate, University of Sheffield, language: English, comment: Part two of the exploration of global class field theory. This contains most of the proofs and applications., abstract: This document is a continuation of my Semester 1 project on class field theory. In the previous work, we made a rounded exposition of the fundamentals of class field theory but in order to preserve the document length the main proofs had to be skipped. We concentrate on filling in the gaps in this second installment. Due to the need to complete the arguments left open last semester and the need for applications this part of the project is a little longer than it should have been. It was not mentioned in the previous project but the class field theory we are studying here is global class field theory. There is such a thing as local class field theory in which we study the Abelian extensions of local fields (essentially fields that arise as completions of a number field with respect to places). Actually we touch on these ideas slightly in this project but never quite get to de_ning a local Artin map and looking at the local analogues of the main theorems of global class field theory. For those wanting to continue on to study local class field theory, consider Chapter 7 of 2] To start off this project we shall first restate the main de_nitions and theorems. This will be brief and those wanting to remind themselves of the details should consult my Semester 1 project. There will be very little motivation or technical results here since this was the purpose of the work done previously. We then set out to prove the main theorems of class field theory. With our present knowledge this would not be a simple task and we soon find that we first have to invent or discover new concepts such as the idele group and the corresponding idele class group. These are topological devices that take stock of all completi

Research Paper 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 DEGREES2 + ny DEGREES2 for integers x, y and posi