Algebraic Number theory (MATH 203a, Fall 2009)

Day & Time: Tuesday and Friday 10:40am — 12:00

Location: Goldsmith (Math) 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.

    • Tentative Schedule of Lectures

    > > > > >
    Class Topic Suggested readings Exercises
    Aug 28 History of Number Theory and presentation of the course [C, chapter 1] Commutative Rings Refresher
    Sept 1 Reminder on trace and norms. Dedekind rings: stability, unique decomposition of ideals [S, chapter 3] or [F, II.1] or [Ma, Chapter 3] or [Mi] or [J] or [L] Exercises 1
    Sept 4 Dedekind rings: unique decomposition of ideals, weak Approximation theorem, Discrete valuation domains [S, chapter 5] or [F. III.1] or [Ma] or [Mi] ot [J] or [L]
    Sept 8 Modules over a Dedekind domain. Decomposition of a prime in an extension. [F, II.4] or [Mi]. Plus [S, chapter V] or [L, I.3] or [Mi] or [Ma,3] Exercises 2
    Sept 11 Discriminant. How to determine the ring of algebraic integers of a number field. The norm of an ideal. Example of quadratic fields, the law of quadratic reciprocity. All the books in the bibliography contain a treatment of this material.
    Sept 15 The Galois case, Frobenius elements. Exercises 3
    Sept 18 Frobenius elements. Cyclotomic fields.
    Sept 22 Proof of the law of quadratic reciprocity. Norm of an ideal. Geometric methods: Lattices and Minkowski theorem. Exercises 4
    Sept 25 Applications of Minkowski theorems and geometric methods : Dirichlet's theorem on the class number. Hermite-Minkowski's theorem; Hermite's theorem; Dirichlet unit's theorem. [S,chapter IV], [Mi, Chapter 4] Write a solution of exercise 7 of set 1, exercises 2 of set 2, and at least six exercises in set 3 belonging to at least three different sections.
    Sept 29 No class (Brandeis' Monday)
    Oct 2 Further geometric methods: Dirichlet's unit theorem. [S, Chapter IV], [Mi, Chapter 5]. Exercises 5
    Oct 6 Dirichlet units theorem. Regulator. Asymptotic estimation of the number of ideals in a class. [L, Chapter VI] Exercises 6
    Oct 9 Analytic methods. Abel's transform. L-functions. [L,VIII.1] or [J,IV] or [Se]
    Oct 13 Analytic methods. The Riemann Zeta function. Euler Product. The Dedekind Zeta funcions of a number fields. The Class Number Formula [L, VIII.1 and VIII.2] or [J,IV] Exercises 7
    Oct 16 Density of a set of primes. The Frobenius Density Theorem. Consequences. [J, IV] for Frobenius and [J, IV] or [L, VIII] or [Se2] for the notions and density
    Oct 20 Dirichlet L-functions and Dirichlet's theorem on prime numbers. [J, IV] for Frobenius and [Se2] or [L, VIII.4] for Dirichlet Exercises 8
    Oct 23 Generalized ideal class groups. Generalized Dirichlet L-functions. The Norm Index Inequality. (End of part I) [L, VI.1 and VIII.3]
    Oct 27 Absolute Value and valuations. [L, II.1] is quick and does not prove Ostrowski. [M, chapter 7] is complete. [Se2] is a good introduction to p-adic numbers.
    Oct 30 Non archimedean absolute values and their rings. Places. Ostrowski's theorem. The Product Formula. [M, Chapter 7]
    Nov 3 Completion. Local fields. [M, Chapter, VII], [L, II]
    Nov 6 Classification of local fields. Hensel's lemma.[M, Chapter VII], [L, II]
    Nov 10 Hensel's Lemma, Krasner Lemma and consequences: unramified extensions, totally ramified extensions, finiteness of the number of extensions of a given degree. The algbraic closure of Q_p is not complete. The field C_p. [M, Chapter VII] Exercises 9
    Nov 13 Places above a given place in an extension of number fields; classification of all places in a number field; the product formula. Product and restricted product. [M, Chapter VII], [F,III.1]. [L, VII.1]
    Nov 17 Adeles. The diagonal embeding. Ideles. [L. VII.2 to VII.5] Exercises 10
    Nov 20 Ideles classes, and generalized ideal class groups. [L,VII.2 to VII.5] Exercises 11
    Nov 24 Class field theory
    Dec 1 Class field theory Exercises 12