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
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