Syllabus

The actual material covered in class can be found in Google Docs. What follows is a tentative syllabus for the class, taken from the Graduate Handbook:

1. Linear and multilinear algebra (4 weeks)
   • Minimal and characteristic polynomials. The Cayley-Hamilton Theorem.
   • Similarity, Jordan normal form and diagonalization.
   • Symmetric and antisymmetric bilinear forms, signature and diagonalization.
   • Tensor products (of modules over commutative rings). Symmetric and exterior algebra (free modules). Hom_R(- , -) and tensor products.

   References: Roman, Chapters 8-10 and 14, Lang, Chapters XV and XVI; Dummit and Foote, Chapter 11.

2. Rudiments of homological algebra (2 weeks)
   • Categories and functors. Products and coproducts. Universal objects, Free objects. Examples and applications.
   • Exact sequences of modules. Injective and projective modules. Hom_R(- , -), for R a commutative ring. Extensions.

   References: Lang, Chapter XX; Dummit and Foote, Part V, Chapter 17.

3. Representation Theory of Finite Groups (2 weeks)
   • Irreducible representations and Schur's Lemma.
   • Characters. Orthogonality. Character table. Complete reducibility for finite groups. Examples.

   References: Lang, Chapter XVIII; Dummit and Foote, Part VI; Serre.

4. Galois Theory (6 weeks)
   • Irreducible polynomials and simple extensions.
   • Existence and uniqueness of splitting fields. Application to construction of finite fields. The Frobenius morphism.
   • Extensions: finite, algebraic, normal, Galois, transcendental.
   • Galois polynomial and group. Fundamental theorem of Galois theory. Fundamental theorem of symmetric functions.
   • Solvability of polynomial equations. Cyclotomic extensions. Ruler and compass constructions.