Introduction to Number Theory, 04 January-26 February 2021

Wednesday, Thursday, Friday 8:00-10:00 (Rio de Janeiro's time)

meet.google.com/vmr-oome-urw

You can access the above link only before 7:55. After this time, you can only follow the course through YouTube:

https://www.youtube.com/impabr

Assistant: Gabriel Fazoli. Register in his list:

https://forms.gle/rT1TFj5VYiwviHyd9

Sometimes he will use the link meet.google.com/ewc-ouuo-nyk for the exposition of exercises.


References




Lectures

  • Week 1: 06-07-08 January.
    1. Chapter 1 [Ireland-Rosen], Section 1,  Unique factorization for integers, page 3,4,5,

    2. Chapter 2, [Ireland-Rosen] Section 1, Theorem 1 (Euclid), there are infinitely many primes in Z. Section 2 Arithmetic functions, page 18,19,20, Moebius inversion theorem. page 21, 22,23,24,25, Section 3,  Theorem 3, sum 1/p diverges, Section 4, Growth of pi(x), page 22,23,24,25,

    3. Exercises: [IrelandRosen] 1.23, 1.25, 1.26, 1.30,  2.4,  2.5,  2.6,  2.10,  2.11, 2.15,  2.26, 2.27.    [Brochero-Moreira-Saldanha-Tengan]  1.1, 1.3, 1.5, 1.12, 1.19, 1.20, 1.21.     

    Lecture 1, Lecture 2
  • Week 2: 13-14-15 January
    1. Chapter 3 [Ireland-Rosen, pages 28-40] Congruence of integers mod n, units and linear equations (prop 3.3.1,3.3.2), Euler and Fermat's Little Theorem, systems of linear equations (Chinese Remainder Theorem, section 4), Wilson's theorem (prop 4.1.1 and Corollary). 

    2. Chapter 4 [Ireland-Rosen, pages 40-45] Roots of unity mod p (prop 4.1.2), primitive roots mod p (theorem 1), primitive roots mod a power of p (theorem 2, 2'), structure of units mod n (theorem 3), classification of numbers admitting primitive roots (prop 4.1.3). 

    3. Exercises:   [IrelandRosen].  3.1, 3.16, 3.21, 3.23, 3.24, 3.25, 3.26, 4.7, 4.15, 4.24.  [Brochero-Moreira-Saldanha-Tengan] 1.33, 1.34, 1.36, 1.46, 1.49, 1.60, 1.77, 1.79, 1.80, 1.82   

  • Week 3: 20-21-22 January.
    1. Chapter 5 [Ireland-Rosen, pages 50-60] Quadratic residues (prop 5.1.1), Legendre symbol and computations (prop 5.1.2, Corollary 3, Gauss' Lemma, prop 5.1.3), Quadratic Reciprocity Law (theorem 1 and proof, section 3).

    2. Chapter 1 [Ireland-Rosen, pages 8-13] The ring of Gauss' integers (prop 1.4.1), Chapter 4 [Garcia-Lequain, pages 101-108] Primes that can be written as sum of two squares (theorem IV.1.2), irreducible elements of Z[i] (Corollary IV.1.3), Fermat's Theorem on numbers that can be written as sum of two squares (theorem IV.1.6), Pythagorean triples (theorem IV.2.2). 

    3. Chapter 4 [Brochero-Moreira-Saldanha-Tengan, pages 126-139] Legendre's Theorem (theorem 4.4), Gauss' Three Squares Theorem (theorem 4.15).

    4. Exercises: [IrelandRosen] 5.3, 5.7, 5.8, 5.11, 5.13, 5.15, 5.16, 5.18, 5.19. [Brochero-Moreira-Saldanha-Tengan] 1.51, 1.52, 1.53, 1.81, 1.83, 2.7, 2.8, 2.12, 2.19, 2.20, 2.22.

  • Week 4: 27-28-29 January
    1. Chapter 5 [Martin, pages 31-35] The ring Z[\sqrt{n}] (Lemma 5.1, 5.2, Bramhagupta's composition rule), units of Z[\sqrt{n}] (Lemma 5.5, theorem 5.6), Dirichlet's aproximation Theorem (prop 5.8), solution to Pell's equation (theorem 5.9), Chapter 4 [Brochero-Moreira-Saldanha-Tengan, pages 140-142] Minkowski's Theorem (theorem 4.18), Sums of four squares (theorem 4.20).

    2. Chapter 17 [Ireland-Rosen, pages 284-288] The ring Z[w] (prop 1.4.2), Fermat's equation for exponent 3 (prop 17.8.1), cubic equation with infinite rational points (prop 17.9.1). 

    3. Chapter 12 [Ireland-Rosen, pages 174-180] Unique factorization rings, Rings of integers of a number field, Dedekind rings and unique factorization of ideals, the class group.

    4.  Chapter 14 [Ireland-Rosen, pages 215-216] Some properties of cyclotomic fields (section 5, Lemma 1, 2), Chapter 1 [Washington, pages 1-7] The First case of Fermat's Last Theorem for regular primes.

    5. Exercises, see the link

  • Week 5: 03-04-05 February
    1. Chapter 6 [Ireland-Rosen], Algebraic numbers, algebraic integers, number field, the set of algebraic integers is a ring, Quadratic Gauss sum, Proof of Reciprocity law using Gauss sum, the sign of the Gauss sum, idea of the proof,
      Exercises: Chapter 6 [Ireland-Rosen], 1,2,4,5,8,10,15,16,17,18,20, 21,
  • Week 6: 10-11-12 February
    1. Chapter 7 [Ireland-Rosen], Finite fields, The multiplicative group of a finite field is cyclic, the existence of finite fields, applications to quadratic residues,
      Chapter 10 Basics of equations over finite fields, Chapter 11 The Zeta function for hypersurfaces over finite fields, examples,
      Exercises: [Hossein Movasati], Chapter 7: 3,4,,5,6,8,9,1014,15, 16, 18, 21,22,23. Chapter 11, Ex. 4 (compute the zeta function of x_0x_1-x_2x_3). Ex. Compute the zeta function of x_0x_1...x_n=0.

  • Week 7: 18-19 February
    1. Week 7, [Hossein Movasati], Basics of complex analysis and holomorphic functions, absolute convergence, Chapter 16 [Ireland-Rosen], Riemann's zeta function, Dirichlet character, Dirichlet L-functions, see also the chapter on Zeta function of my lecture notes
      Exercises: Chapter 16, [Ireland-Rosen] Ex. 2,3,4,5,6,8,9,10,13,14,15.

  • Week 8: 24-25-26 February

  • Special problems arising from Complex Geometry and Hodge Theory. 

    Any solution, or any effort to find a solution, will contribute to improve your grade (C can trun into B and B into A).

    1. Problem 1, Configuration of lines

    2. Problem 2, Differential equations and number theory

    3. Problem 3, A big matrix: How to compute its rank? I

    4. Problem 4.  A big matrix:How to compute its rank? II