McGill University
Department of Mathematics & Statistics
Number Theory
189346A / 377B
Detailed Syllabus
(The chapter numbers refer to Leveque, "Fundamentals of Number Theory").
 Jan 3Jan 5: (Chapters 1,2). Overview of the course. Basic
properties of the integers. The GCD and the Euclidean algorithm.
Proof of the fundamental theorem of arithmetic. Application to
Diophantine equations.

Jan 8Jan 12: (Chapter 2).
Modular arithmetic. Wilson's Theorem.
Fermat's Little Theorem.
Primality testing and factorisation.
 Jan 15Jan 19: (Sections 3.1, 3.2, 4.1, 4.2).
The structure of (Z/nZ)^{x}. The Euler phifunction.
Application to primality testing and cryptography.
Congruence equations. The Chinese remainder theorem.
Structure of the polynomial ring $Z/pZ[x]$.
 Jan 22Jan 26:
Congruence equations, continued.
padic numbers.
 Jan 29 Feb 2 : (Chapter 4,5).
Discrete logarithms. The DiffieHellman key exchange.
Power residues.
 Feb 5  Feb 9:
Midterm exam on Monday.
The padic logarithm, and some review.
 Feb 12  Feb 16: (Chapter 6).
The law of quadratic reciprocity.
 Feb 19 Feb 23:
Study break. A good time to work seriously on
your project! In particular, your topic should have been chosen by then.
 Feb 26  March 2 :
(Chapter 6). Introduction to analytic number theory.
Euler's proof of the infinitude of primes.
The sieve of Eratosthenes.
Dirichlet's theorem on primes in arithmetic progressions.
 March 5  March 9: (Chapter 6).
Dirichlet's Theorem, continued.
 March 12  March 16:
(Sec. 2.2 and Chapter 8).
Quadratic equations and
quadratic fields: Legendre's Theorem.
 March 19  March 23:
(Chapter 7) Sums of squares.
 March 26  March 30: (Chapter 8, 9).
Diophantine approximation and continued fractions.
 April 2  April 4: Continued fractions and Pell's equation
(Chapter 9)
 April 11: Review.