Advanced Algebra

The course is mainly devoted to the classical algebraic structures considered from the modern point of view. The categorical language is developed in the amount sufficient for considering the categories of groups, rings, modules, field extensions, etc., as well as for discussing the functors relating to them. The problem of classification up to isomorphisms is going the be the central one. Special attention will be paid to the structures related to the classical arithmetic.

Prerequisites: Basic Algebra and Linear Algebra.


Curriculum:

  1. Categories and functors (language, overview).
  2. Foundational issues. Small and essentially small categories.
  3. Classification problems and representable functors.
  4. Additive and abelian categories. Derived functors.
  5. Group objects in categories.
  6. Groups and their extensions.
  7. Cohomology of groups. Interpretation in small (0,1,2,3) dimensions.
  8. Simple (finite and Lie) groups.
  9. Galois correspondence.
  10. Categories of extensions of the ground field.
  11. Galois theory of field extensions.
  12. Algebraic closure of a field. Separable closure.
  13. Finite fields and their algebraic closures.
  14. Number fields. The absolute Galois group.
  15. Galois cohomology.
  16. Finitely-generated extensions of an algebraically closed field.
  17. Transcendence degree of a field extension. Birational geometry.
  18. Abelian extensions of number fields  and functional fields.

Textbooks

  • D.S.Dummit, R.M.Foote, Abstract Algebra.
  • Robert B.Ash, Abstract Algebra: The Basic Graduate Year.