Algebra I (MAT 534, Fall semester)

1. Groups (5 weeks)
   • Direct products, Normal subgroups, Quotient groups, and the isomorphism theorems.
   • Groups acting on sets; orbits and stabilizers. Applications: class formula, centralizers and normalizers, centers of finite p-groups. Conjugacy classes of S_n
   • Sylow's Theorems, Solvable groups, Simple groups, simplicity of A_n. Examples: Finite groups of small order ($\le$8).
   • Structure of finitely generated abelian groups. Free groups. Applications.

   References: Lang, Chapter I; Dummit and Foote, Part I; Rotman.

2. Basic linear algebra (3 weeks)
   • Vector spaces, Linear dependence/independence, Bases, Matrices and linear maps. Dual vector space, quotient vector spaces, isomorphism theorems.
   • Determinants, basic properties. Eigenspaces and eigenvectors, characteristic polynomial.
   • Inner products and orthonormal sets. Spectral theorem for normal operators (finite dimensional case).

   References: Lang, Chapters XIII and XIV; Dummit and Foote, Chapter 11.

3. Rings, modules and algebras (6 weeks)
   • Rings, subrings, fields, ideals, homomorphisms, isomorphism theorems, polynomial rings.
   • Integral domains, Euclidean domains, PID's. UFD's and Gauss's Lemma ( F[x₁,..., x_n] is an UFD). Examples.
   • Prime ideals, maximal ideals. The Chinese remainder Theorem. Fields of fractions.
   • The Wedderburn Theorem (no proof). Simplicity and semisimplicity.
   • Noetherian rings and the Hilbert Basis Theorem.
   • Finitely generated modules over PID's, the structure theorem.

   References: Lang, Chapters II, III, V, and VI; Jacobson, Chapter 2; Dummit and Foote, Part II.

Typical References:

• S. Lang, Algebra,
  3^rd ed., Addison-Wesley, 1993.

• Jacobson, Basic Algebra,
  2^nd ed, W.H. Freeman, New York, 1985, 1989.

• Dummit and Foote, Abstract Algebra,
  2^nd ed, John Wiley, 1999.

• Rotman, Introduction to the Theory of Groups,
  Springer Verlag.