Syllabus
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:
 Linear and multilinear algebra (4 weeks)
 Minimal and characteristic polynomials. The CayleyHamilton 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 810 and 14, Lang, Chapters XV and XVI; 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.
Hom_{R}( , ), for R a commutative ring.
Extensions.
References: Lang, Chapter XX; Dummit and Foote, Part V, Chapter 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 XVIII; 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,
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.
