Timing: Tuesdays 1 pm to 2 pm, Thursdays 4 pm to 5 pm, Fridays 4 pm to 5 pm starting 29 Jan
Room: PHYS 3.21 on Tuesdays, MATH SM2 on Thursdays and Fridays
Office hours: Thursdays 2 pm to 3 pm, or by appointment
Content
The aim of this course is to enable students to gain an understanding and appreciation of algebraic number theory and familiarity with the basic objects of study, namely number fields and their rings of integers. In particular, it should enable them to become comfortable working with the basic algebraic concepts involved, to appreciate the failure of unique factorisation in general, and to see applications of the theory to Diophantine equations.
More precisely, we will cover the following topics:
Review of Integral domains, UFD, PID, Euclidean Domains
Gaussian integers, application to writing primes as sums of two squares
Algebraic numbers and algebraic integers
Basics of field extensions and embeddings of number fields in complex numbers
Norms, traces and discriminants
Integral bases of number rings
Dedekind domains and unique factorization for Dedekind domains
Norms of ideals
Splitting of primes in extensions
Case study 1: quadratic fields
Case study 2: cyclotomic fields
The Minkowski bound for lattices
The ideal class group and finiteness of ideal class group
The Dirichlet unit theorem
Application to Pell-type equations and some equations of the type y^2 = x^3 + k
Prerequisites
Algebra 2 (Math 21800). You should be comfortable with the basics of rings and fields.
Reading
I. Stewart and D. Tall: Algebraic Number Theory and Fermat’s Last Theorem (Relevant chapters are 1 to 9, and Appendix B)
D. Marcus: Number fields (This is the most relevant book for this course: the relevant chapters are 2,3,5 and Appendices 1 and 2)
Esmonde and Murty: Problems in algebraic number theory (E-book available from the library. Relevant chapters are 2,3,4,5,6, 8)
S. Alaca and K.S. Williams: Introductory Algebraic Number Theory (All Chapters are relevant)
Lang: Algebra (Relevant sections: Chapter 5, Section 1 to 4. Note that this is only for supplementary reading; you will not be examined on any material here that is not otherwise covered in the notes.)
