Algebra I (MATH 101a, Fall 2013)



Day & Time: Tuesday and Thursday 3:30pm — 4:50am

First day of class: Tuesday, September 3

Location: Goldsmith 116

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

  • Groups; Morphisms of 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 examples 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

    Class Topic Readings Exercises
    Sept 3 Presentation of the course. Groups: definitions; subgroups; subgroup generated by a subset; morphisms;
    Sept 10 Groups: cosets; normal subgroups;
    Sept 12 Groups: factor groups; canonical isomorphisms; Tower of groups; Schreier's theorem Exercises
    Sept 17 Groups: Jordan-Holder theorem; extensions; direct and semi-direct product;
    Sept 24 Semi-direct product and split extensions
    Oct 1 Groups: cyclic and dihedral groups; order of an element;
    Oct 3 Torsion-free and free abelian groups
    Oct 8 Finitely generated abelian groups. Exercises Solutions
    Oct 10 Groups: Operations of a group on a set;
    Oct 15 Symmetric groups; alternating groups;
    Oct 17 Alternating groups;
    Oct 22 Simplicity of the alternating group. finite groups (Lagrange's theorem); Exercises
    Oct 24 Sylow's theorems and applications.
    Oct 29 Categories. Diagrams. Initial and final objects.
    Oct 31 Categories: limits of diagrams Exercises
    Nov 5 Categories: monomorphisms and epimorphisms. Functors. Natural Transformations
    Nov 7 Rings, domains, fields: definitions
    Nov 12 Rings: ideals; principal ideal domain; noetherian rings;
    Nov 14 Rings: prime and maximal ideal; irreducible elements; UFD's; ExercisesSolutions
    Nov 19 Rings: rings of polynomials; Hilbert's basis theorem;
    Nov 21 Rings: fraction field of a domain; field of rational functions;
    Nov 25 (Monday) Fields: extensions of field; degree of a finite extension; multiplicativity of degree;
    Nov 26 Fields: algebraic and transcendental extensions; Exercises
    Dec 3 Fields: normal and separable extensions;
    Dec 5 Fields: Galois extensions and Galois groups; Galois theory; Exercises
    Supplementary session (Dec 10, 3pm) Fields: Applications of Galois theory to the solution of algebraic equations and to geometric constructions. Exercises(not due)