McGill University
Department of Mathematics & Statistics
Number Theory
189346A / 377B
Detailed Syllabus
(The chapter numbers refer to the texts by
Granville and by Leveque which will be our basic references.)
 Jan 7Jan 11: (Levesque, Chapter 1).
A bit of overview of the course. Remarks about number systems (Integers, rational numbers,
real numbers, complex numbers...) Algebraic and transcendental
numbers. Number fields and their rings of integers.

Jan 14Jan 18: (Granville, Chapter 1; Levesque, Chapter 2).
Unique factorisation and the Euclidean algorithm.
Proof of the fundamental theorem of arithmetic and its
extnesion to Euclidean domains. Arithmetic application
of unique factorisation:
Diophantine equations and expressing integers as sums of squares.
 Jan 21Jan 25: (Granville, Chapters 2 and 4;
Levesque, Sections 3.13.4.)
Unique factorisation and the Euclidean algorithm, cont'd.
Modular arithmetic.
 Jan 28Feb 1:
Wilson's Theorem and
Fermat's Little Theorem.
The structure of (Z/nZ)^{x}. The Euler phifunction.
Congruence equations. Hensel's Lemma, and the Chinese remainder theorem.
 Feb 4 Feb 8 : (Granville, Chapter 7;
Levesque, Chapters 4 and 5).
Primality testing and factorisation.
Application to cryptography.
The RSA public key cryptosystem.
 Feb 11  Feb 15:
Discrete logarithms.
The DiffieHellman key exchange.
The mod p^n logarithm.
 Feb 18  Feb 22: (Levesque, Chapter 3,4).
Review on Monday.
padic numbers. padic logarithms.
Hensel's lemma, revisited.
 Feb 25  March 1 :
(Levesque, Chapter 5).
On Monday, February 25, there will be the Midterm exam.
The law of quadratic reciprocity.
 March 4 March 8:
Study break. A good time to work seriously on
your project! In particular, your topic should have been chosen by then.
 March 11  March 15 :
Quadratic reciprocity, cont'd.
 March 18  March 22:
(Granville, Chapter 5 and Levesque, 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 25  March 29:
(Levesque, Chapter 6).
Dirichlet's Theorem, continued.
 April 1  April 5:
(Granville, Sec. 1.3. and Chapter 11 and Levesque, Chapters 8, 9).
Pell's equation, rudiments of diophantine approximation,
Continued fractions.
 April 8  April 12:
(Granville, Sec. 1.3. and Chapter 11 and Levesque, Chapters 8, 9).
Pell's equation, rudiments of diophantine approximation,
Continued fractions, cont'd.
 April 15: Review.