MAU23101 - Introduction to number theory

Course information

Suggested textbooks

• A course in computational number theory, by D. Bressoud and S. Wagon
• The higher arithmetic, by H. Davenport (up to chapter VI)
• A classical introduction to modern number theory, by K. Ireland and M. Rosen (up to chapter 5)
• Primes of the form x²+ny², by D. Cox (up to section 2.B)

Recorded lectures

• Chapter 0: Introduction - Diophantine equations
  Video - Notes


• Chapter 1: Divisibility and factorisation
  Notes
  Videos:
  • Euclidean division (17:16)
  • Divisibility, gcd, lcm (24:09)
  • The fundamental theorem of arithmetic (11:43)
  • Practical factoring (08:24)
  • Valuations (13:04)
  • Multiplicative functions (14:38)
  • The Diophantine equation ax+by=c (19:46)
  • Complements on primes (12:00)


• Chapter 2: Congruences and Z/nZ
  Notes
  Videos:
  • The set Z/nZ (08:38)
  • The ring Z/nZ (15:18)
  • Application to Diophantine equations (13:25)
  • Invertible elements (32:39)
  • Chinese remainders (28:20)
  • Additive and mutiplicative order (25:52)
  • Primitive roots (54:44)


• Chapter 3: Power residues, Legendre symbols, and quadratic reciprocity
  Notes
  Videos:
  • Powers in Z/pZ (24:36)
  • The Legendre symbol (21:22)
  • Proofs part 1/3 (08:28)
  • Proofs part 2/3 (18:00)
  • Proofs part 3/3: quadratic reciprocity (17:31)


• Chapter 4: Sums of squares
  Notes
  Videos:
  • Introduction (10:46)
  • The Gaussian integers (11:29)
  • Arithmetic with the Gaussian integers (35:31)
  • Classification of the Gaussian primes (40:17)
  • Conclusion (20:01)


• Chapter 5: Binary quadratic forms
  We will probably not have time to treat this chapter in class. Here are the lecture slides and videos, in case you would still like to take a look at them.
  Notes
  Videos:
  • Introduction, representation by a form (07:23)
  • Equivalent forms, discriminant (19:48)
  • Equivalence vs. representability (14:35)
  • Reduced forms (48:55)
  • The class number (10:24)
  • The class number one case (09:35)


• Chapter 6: Continued fractions
  Notes
  Videos:
  • Concept and notations (04:54)
  • Continued fraction expansion of a real number (13:36)
  • Convergents (21:13)
  • Convergence of continued fractions (15:15)
  • Diophantine approximation (37:58)
  • Continued fractions attached to quadratic irrationals (47:13)
  • The Pell-Fermat equation (31:12)

Homework

Exam