McGill University
Department of Mathematics & Statistics
Number Theory
189-346A / 377B
Detailed Syllabus
(The chapter numbers refer to the texts by
Granville and by Leveque which will be our basic references.)
- Jan 7-Jan 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 14-Jan 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 21-Jan 25: (Granville, Chapters 2 and 4;
Levesque, Sections 3.1-3.4.)
Unique factorisation and the Euclidean algorithm, cont'd.
Modular arithmetic.
- Jan 28-Feb 1:
Wilson's Theorem and
Fermat's Little Theorem.
The structure of (Z/nZ)x. The Euler phi-function.
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 Diffie-Hellman key exchange.
The mod p^n logarithm.
- Feb 18 - Feb 22: (Levesque, Chapter 3,4).
Review on Monday.
p-adic numbers. p-adic 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.