Algebraic Number Theory (MATH 203a, Fall 2013)



Day & Time: Tuesday and Thursday 2:00pm — 3:20am

First day of class: Tuesday, September 3

Location: Goldsmith 116

Instructor: Joel Bellaiche

Contents : Dedekind domains: definition, stability by integral closure, unique factorization of ideals, modules over Dedekind Domain; Extensions of Dedekind domains: discriminant, behaviour of primes; Ring of algebraic numbers of a number field; Finiteness of the class number; Dirichlet units Theorem; Cyclotomic Extensions; Local Fields; Adeles and Ideles, Grossencharacters; Class field theory; Cebotarev Density Theorem;

Program:

  • PART I: Global Theory
  • Algebraic methods: General theory of Dedekind domains and application to algebraic numbers (3 weeks)
  • Geometric methods: Bounds on discriminants and finiteness results (Class groups, Dirichlet's Unit Theorem) (1 week and a half)
  • Analytic methods: Zeta and L-functions and applications (Class Number Formula, Dirichlet's prime number theorem, Cebotarev) (2 weeks)

  • PART II: Local Theory
  • Complete fields, $p$-adic fields, completions of number fields (1 week)
  • Adeles and Ideles (1 week)

  • PART III: Class Field Theory (local and global)
  • Class Field Theory: adelic formulation of global class field theory; local class field theory; consequences (1 week)
  • Class Field theory: proof of global class field theory; proof of local clas field theory by global methods. (1 week)

    Bibliography : We will not follow one specific book. Good references on algebraic number theory abound. Here are some:

  • [C] Cox. David A, Primes of the form x^2 + n y^2 (cover tangentially and provides arithmetic motivation for some material)
  • [F] Frohlich, Algebraic Number Theory
  • [J] Janusz, Algebraic Number Fields
  • [L], Lang, Serge, Algebraic Number Theory (cover a lot of material quickly, maybe hard to read)
  • [Ma] Marcus, Daniel A, Number Fields (nice, probably a little bit too elementary for this class, contains a lot of exercises)
  • [Mi] Milne, J. S., Algebraic Number Theory, avalaible on the author's web page (contains a good part of the material covered in this course)
  • [N1] Neukirch, Algebaric Number theory
  • [N2] Neukirch, Class Field Theory
  • [S], Samuel, Pierre, Theorie Algebrique des Nombres or Algebraic Number Theory (elementary and efficient coverage of part I. Contains some nice exercises)
  • [Se] Serre, Jean-Pierre, Local Fields (cover algebraic methods in part I, all of part II, and local class field theory with a purely local proof)
  • [Se2] Serre, Jean-Pierre, A course in Arithmetic (contains a nice introduction to local fields - and many other things of interest that are beyond the scope of this course).

    You should give a look to most of those books during the first weeks of this course, and then pick up the two or three you like most and stick to them.

    If you need some rest but wish to stay in a number theoretic world, try the novel: Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis.


  • Schedule of Lectures

    Class Topic Readings Exercises
    Sept 3 Presentation of the course. Overview of Number Theory. Reminder on traces and Norms [C, chapter 1] Refresher in algebra
    Sept 10 Reminder on trace and norms. [S, chapter 3] or [F, II.1] or [Ma, Chapter 3] or [Mi] or [J] or [L]
    Sept 12 The Pell-Fermat equation and quadratic fields. [S, chapter 5] or [F. III.1] or [Ma] or [Mi] ot [J] or [L]
    Sept 17 Dedekind domains: Definition, stability
    Sept 24 Dedekind domains: unique decomposition of ideals, weak Approximation theorem.
    Oct 1 Discrete valuation domain. Decomposition of a prime in an extension. Exercises set 1
    Oct 3 Decomposition of a prime in an extension
    Oct 8 Exercises set 2
    Oct 10 Discriminant.
    Oct 15 Decomposition of a prime in a Galois extension.
    Oct 17 Decomposition groups and their structures. Frobenius elements Exercises set 3
    Oct 22 Cyclotomic fields and the law of quadratic reciprocity
    Oct 24 Geometric methods: discrete subgroup of euclidean spaces
    Oct 29 Geometric methods: Finiteness of the class number
    Oct 31 Geometric methods: Finiteness of the set of number fields with a bounded discriminant.
    Nov 5 Geometric methods: Dirichlet's unit theorem
    Nov 7 Analytic methods: L-functions
    Nov 12 Exercises set 4
    Nov 14
    Nov 19
    Nov 21
    Nov 25 (Monday)
    Nov 26
    Dec 3
    Dec 5