- Linear and multilinear algebra (4 weeks)
- Minimal and characteristic polynomials. The Cayley-Hamilton Theorem.
- Similarity, Jor`dan normal form and diagonalization.
- Symmetric and antisymmetric bilinear forms, signature and
- Tensor products (of modules over commutative rings). Symmetric
and exterior algebra (free modules).
HomR(- , -) and
References: Lang, chapters XIII and XIV; Dummit and Foote, Chapter 11.
- 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.
HomR(- , -), for R a commutative ring. Extensions.
References: Lang, chapter XX; Dummit and Foote, Part V, 17.
- 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 XVII; Dummit and Foote, Part VI; Serre.
- 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,
- Galois polynomial and group. Fundamental theorem of Galois
theory. Fundamental theorem of symmetric functions.
- Solvability of polynomial equations. Cyclotomic
extensions. Ruler and compass constructions