Algebra I (MATH 101a, Spring 2008)

Day & Time: Tuesday and Friday 9:10am — 1 :25am

Location: Goldsmith 0226

Instructor: Joel Bellaiche

Textbooks: For references, I will essentially use Algebra of Serge Lang (third or fourth edition -- they are almost identical). Occasionally I will refer to Basic Algebra, of Jacobson. Both books are very good, but they have a different style : Lang is quick and sometimes allusive, while Jacobson is more thorough. I will also use another book for category theory, to be precised later.

Program:

PART I : Groups

  • Monoids; Groups; Morphisms of monoids and groups; subgroups; generating family; cosets;
  • Normal subgroups; Quotient groups; Simple groups; Jordan-Holder theorem; Direct and semi-direct products; extensions;
  • Cyclic groups; finite abelian groups;
  • Operation of a group on a set;
  • Symmetric and alternate groups: definitions, generating familes, simplicity.
  • Solvable groups; p-groups; Sylow subgroups and Sylow theorems;
  • Other example of finite simple groups;

  • PART II : Categories
  • Definition of a category; Examples;
  • Fonctors; Natural transformation of functors; Equivalence of categories;
  • Initial anf final objects of a cateory;
  • Commutative diagrams; limits and colimits; Fibred products and coproducts;
  • Adjoint functors; examples;

  • PART III : Commutative rings
  • Rings, ideals, modules;
  • Principal rings; Factorial rings; Noetherian rings;
  • Local rings and localization; Nakayama's lemma;
  • Projective modules; Free modules;
  • Modules over a principal rings
  • Tensor product of modules; flat modules;

  • PART IV : Fields and Galois theory
  • Fields; Extensions of fields; finite and algebraic extensions;
  • Normal and separable extensions; Perfect fields; Primitive element theorem;
  • Galois theory (standard forms); Solvability of equations;
  • Other formulations of Galois theory;

    Part V : Non commutative ring theory and representations of finite groups;

    • 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