MA2316: Introduction to Number Theory

Hilary term, 2013

Lecturer: Masha Vlasenko

The goal of this course is to introduce the basic concepts of number theory, both analytic and algebraic. We begin with the classical questions such as factorisation, distribution of primes and finding integer solutions to algebraic equations. The second part of the course deals with analytic methods. We will study Riemann zeta function, discuss Riemann hypothesis and prove the theorem on primes in arithmetic progressions. If time admits we will introduce algebraic field extensions and basics of Galois theory, and see that the theorem on primes in arithmetic progressions follows from Chebotarev's density theorem.

The course will be accompanied by weekly tutorials in the form of problem-solving sessions. We will learn how to use computers to do number theory. To follow the computational part of the course please install PARI/GP calculator (or learn how to run it in the computer lab) and have the reference card of PARI functions in hand. Tutorials take place on Thursdays at 2 pm in Syn2. For those who can not attend on Thursdays there is an additional class, which is at the moment on Fridays at 4pm in EELT2.
Prerequisites: basics of analysis, linear algebra and group theory (MA1124,MA1212, MA1214). Knowledge of fields, rings and modules (MA2215) and complex analysis (MA2325) is not necessary but will help to understand selected topics.

Literature
[1] I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, Fifth Edition
[2] J.-P. Serre, A Course in Arithmetic
[3] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory
[4] S. Lang, Algebra

Our major textbook is [1]. This book, although 20 years old, is renowned for its interesting, and sometimes challenging, problems. Please see Montgomery's home page for the lists of typos and errors in the book. Note that there are multiple lists because the book has been reprinted several times.

Comments on the exam paper <here>.

Topics of lectures and relevant material
  1. Introduction: Diophantine equations, Pythagorean triples, Fermat's Last Theorem and the Birch and Swinnerton-Dyer Conjecture. <pdf>
  2. Divisibility and the Euclidean algorithm. <pdf>
  3. Primes, uniqueness of factorisation and integers in quadratic fields. <pdf> <pdf> <pdf><pdf>
  4. Congruences I: Fermat's Little Theorem and Euler's generalisation <pdf>, the Chinese Remainder Theorem <pdf>, prime power moduli<pdf>.
  5. Congruences II: primitive roots <pdf> <pdf>
  6. Congruences III: Legendre symbols and the Quadratic Reciprocity Law.<pdf>
  7. Analytic methods I: Riemann's zeta function.<pdf>
  8. Analytic methods II: Dirichlet's characters, L-functions and primes in arithmetic progressions.<pdf>
  9. Analytic methods III: Riemann's Hypothesis (<pdf>, <pdf>) and distribution of primes <pdf>.
Grading
2 hour examination in May, the final mark will be 80% of the exam mark + 20% of the continuous assessment mark. The continuous assessment consists of two kinds of work:
In case you've got more then 20% in the continuous assessment by the end of the course, your mark will be cut down to 20%. One person can be a reporter only once.The questions on the final exam will be close to the ones in the assignments, but also there will be some theoretical questions.