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 5Jan 7: (Levesque, Chapter 1).
Overview of the course. Remarks about number systems (Integers, rational numbers,
real numbers, complex numbers...) Cardano's solution of the cubic as a motivation for
complex numbers.

Jan 10Jan 14: (Granville, Chapter 1; Levesque, Chapter 2).
Basic properties of the integers. The GCD and the Euclidean algorithm.
Proof of the fundamental theorem of arithmetic. Application of unique factorisation to
some Diophantine equations. First notions concerning congruences.
 Jan 17Jan 21: (Granville, Chapters 2 and 4;
Levesque, Sections 3.13.4.)
Modular arithmetic. Wilson's Theorem and
Fermat's Little Theorem.
The structure of (Z/nZ)^{x}. The Euler phifunction.
Congruence equations. The Chinese remainder theorem.
 Jan 24Jan 28:
Primality testing and factorisation.
Application to cryptography.
Structure of the polynomial ring $Z/pZ[x]$.
padic numbers.
 Jan 31 Feb 4 : (Granville, Chapter 7;
Levesque, Chapters 4 and 5).
Discrete logarithms. The DiffieHellman key exchange.
Power residues.
 Feb 7  Feb 11:
The padic logarithm, and some review.
Midterm exam on Friday.
Here is a practice midterm to help you in your studying.
 Feb 14  Feb 18: (Granville, Chapter 8; Levesque, Chapter 6).
The law of quadratic reciprocity.
 Feb 21 Feb 25:
Study break. A good time to work seriously on
your project! In particular, your topic should have been chosen by then.
 Feb 28  March 4 :
(Granville, Chapter 8 and Levesque, Chapter 6).
The law of quadratic reciprocity.
 March 7  March 11 :
(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 14  March 18: (Levesque, Chapter 6).
Dirichlet's Theorem, continued.
 March 21  March 25:
(Granville, Chapter 12; Levesque, Sec. 2.2 and Chapter 8).
Quadratic fields and quadratic rings.
Unique factorisation, revisited.
 March 28  April 1:
(Granville, Chapter 9; Levesque, Chapter 7).
Sums of squares.
 April 4  April 6:
(Granville, Sec. 1.3. and Chapter 11 and Levesque, Chapters 8, 9).
Pell's equation, rudiments of diophantine approximation,
Continued fractions.
Supplementary reading: Here is a
historical account of
Pell's equation written by Joshua Aaron.
 April 8: Review.