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 5-Jan 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.

  2. Jan 10-Jan 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.

  3. Jan 17-Jan 21: (Granville, Chapters 2 and 4; Levesque, Sections 3.1-3.4.)
    Modular arithmetic. Wilson's Theorem and Fermat's Little Theorem. The structure of (Z/nZ)x. The Euler phi-function. Congruence equations. The Chinese remainder theorem.

  4. Jan 24-Jan 28: Primality testing and factorisation. Application to cryptography. Structure of the polynomial ring $Z/pZ[x]$. p-adic numbers.

  5. Jan 31- Feb 4 : (Granville, Chapter 7; Levesque, Chapters 4 and 5).
    Discrete logarithms. The Diffie-Hellman key exchange. Power residues.

  6. Feb 7 - Feb 11: The p-adic logarithm, and some review. Midterm exam on Friday.
    Here is a practice midterm to help you in your studying.

  7. Feb 14 - Feb 18: (Granville, Chapter 8; Levesque, Chapter 6).
    The law of quadratic reciprocity.

  8. 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.

  9. Feb 28 - March 4 : (Granville, Chapter 8 and Levesque, Chapter 6).
    The law of quadratic reciprocity.

  10. 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.

  11. March 14 - March 18: (Levesque, Chapter 6).
    Dirichlet's Theorem, continued.

  12. March 21 - March 25: (Granville, Chapter 12; Levesque, Sec. 2.2 and Chapter 8).
    Quadratic fields and quadratic rings. Unique factorisation, revisited.

  13. March 28 - April 1: (Granville, Chapter 9; Levesque, Chapter 7).
    Sums of squares.

  14. 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.

  15. April 8: Review.