SUNY at Stony Brook MAT 535: Algebra II
Fall 2016


What follows is a tentative syllabus for the class, taken from the Graduate Handbook.

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). HomR(- , -) 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. HomR(- , -), 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.