Algebraic Number theory (MATH 203a, Spring 2008)

Day & Time: Monday and Wednesday 2:10am — 3:25am

Location: Goldsmith 0226

Instructor: Joel Bellaiche

Contents : Dedekind domains: definition, stability by integral closure, unique factorization of ideals, modules over Dedekind Domain; Extensions of Dedekind domain : 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, Grossencharacter; Class field theory; Cebotarev Density Theorem; Galois representations

Tentative program:

  • PART I : Global Theory
  • (Algebraic methods) : General theory of Dedekind domains and application to algebraic numbers (2 weeks)
  • (Geometric methods) : Bounds on discriminants and finiteness results (Class groups, Dirichlet's Unit Theorem) (2 weeks)
  • (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
  • Class Field Theory : adelic formulation and consequences (1 week and half)
  • Class Field theory : proofs or skecths of the proofs (1 week)

  • PART IV : Introduction to the theory of Galois representations
  • Grossenchracters and algebraic grossenchracters (1 week)
  • Higher dimensional Galois representations (depending on time)

    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 Field
  • [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
  • [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 the first third of the material of this class. Includes some nice exercises)
  • [Se] Serre, Jean-Pierre, Local Fields
  • [Se2] Serre, Jean-Pierre, A course in Arithmetic

    You should give a look to most of those books during the first weeks of this course, and then pick up the two 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

    > > > > > [M, CHapter VII], [L, II]
    Class Topic Suggested readings Exercises
    Jan 16 Introduction. History of number theory. [C, chapter 1] Commutative Rings Refresher (new version)
    Jan 23 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
    Jan 28 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]
    Jan 30 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
    Feb 04 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.
    Feb 06 The Galois case, Frobenius elements. Exercises 3
    Feb 11 Frobenius elements. Cyclotomic fields.
    Feb 13 Proof of the law of quadratic reciprocity. Norm of an ideal. Geometric methods : Lattices and Minkowski theorem. Exercises 4
    Feb 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] Due : Set 1: exercises 3 and 7. Set 2: exercise 5. Set 3, section 2: exercises 1 to 6.
    Feb 27 Further geometric methods: Dirichlet's unit theorem. [S, Chapter IV], [Mi, Chapter 5]. Exercises 5
    Mar 3 Dirichlet units theorem. Regulator. Asymptotic estimation of the number of ideals in a class. [L, Chapter VI] Exercises 6
    Mar 5 Analytic methods. Abel's transform. L-functions. [L,VIII.1] or [J,IV] or [Se]
    Mar 10 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
    Mar 12 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
    Mar 17 Dirichlet L-functions and Dirichlet's theorem on prime numbers. [J, IV] for Frobenius and [Se2] or [L, VIII.4] for Dirichlet Exercises 8
    Mar 19 Generalized ideal class groups. Generalized Dirichlet L-functions. The Norm Index Inequality. (End of part I) [L, VI.1 and VIII.3]
    Mar 24 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.
    Mar 26 No class this day. Sorry
    Mar 31 Non archimedean absolute values and their rings. Places. Ostrowski's theorem. The Product Formula. [M, Chapter 7]
    April 2 Completion. Local fields. [M, Chapter, VII], [L, II]
    April 7 Classification of local fields. Hensel's lemma.
    April 9 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 (some typos corrected)
    April 14 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]
    April 16 Adeles. The diagonal embeding. Ideles. [L. VII.2 to VII.5] Exercises 10
    April 28 Ideles classes, and generalized ideal class groups. [L,VII.2 to VII.5] Exercises 11 Due: Exercises 1 and 2 of section 3 of exercises set 8. Questions a,b,c of exercise 7 of exercises set 9.
    April 29 (4pm--5:20pm) Class field theory
    April 30 Class field theory Exercises 12