Algebraic Number Theory, Polygons and Quadratic Reciprocity
This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan. This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan.
Class Field Theory
The first part of my masters dissertation, completed under the supervision of Dr. Neil Dummigan.
This is... more
The first part of my masters dissertation, completed under the supervision of Dr. Neil Dummigan.
This is a quite informal view of global class field theory, viewed from the platform of ideals.
See the second part, "Class Field Theory: Proofs and Applications", for a more detailed view along with proofs, including the introduction of ideles, a bit of cohomology and applications of class field theory to the representation of primes by the quadratic form x^2 + ny^2.
Class Field Theory: Proofs and Applications
The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan.
This... more
The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan.
This installment proves everything done informally in the first part. This is quite a difficult and lengthy task and many new devices need to be invented, such as the ideles and the Herbrand quotient.
Finally, we apply the theory to the representation of primes by the quadratic form x^2 + ny^2, giving some examples.
