Introduction to Number Theory, January-February 2017.
Important dates:



  1.   Week 1,[Marcus Torres],     

  2. Week 2, [Roberto Villaflor]  

  3. Week 3,  [Roberto Villaflor],                           

  4. Week 4, [Roberto Villaflor],

  5. Week 5, [Hossein Movasati], 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,

  6. Week 6, [Hossein Movasati], 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.

  7. 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.

  8. Week 8, [Hossein Movasati], Two lectures on elliptic curves, see my lecture notes

4 References:
[Ireland-Rosen] "A Classical Introduction to Modern Number Theory", Second edition 1990. (For exercises notice the edition)
[Brochero-Moreira-Saldanha-Tengan] "Teoria dos Números: um passeio com primos e outros números familiares pelo mundo inteiro", 4th edition
[Garcia-Lequain] "Elementos de Álgebra"
[Martin] "Introduction to Number Theory"
[Washington] "Introduction to Cyclotomic Fields"

4 Exercises 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