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.)
  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. Feb 4- Feb 8 : (Granville, Chapter 7; Levesque, Chapters 4 and 5).
    Primality testing and factorisation. Application to cryptography. The RSA public key cryptosystem.

  6. Feb 11 - Feb 15: Discrete logarithms. The Diffie-Hellman key exchange. The mod p^n logarithm.

  7. Feb 18 - Feb 22: (Levesque, Chapter 3,4).
    Review on Monday.
    p-adic numbers. p-adic logarithms. Hensel's lemma, revisited.

  8. Feb 25 - March 1 : (Levesque, Chapter 5).
    On Monday, February 25, there will be the Midterm exam.
    The law of quadratic reciprocity.

  9. 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.

  10. March 11 - March 15 : Quadratic reciprocity, cont'd.

  11. 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.

  12. March 25 - March 29: (Levesque, Chapter 6).
    Dirichlet's Theorem, continued.

  13. 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.

  14. 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.

  15. April 15: Review.