Syllabus
What follows is a tentative syllabus for the class, taken from the Graduate Handbook:
 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 pgroups. Conjugacy classes of S_{n}
 Sylow's Theorems, Solvable groups, Simple groups, simplicity of A_{n}. Examples: Finite groups of small
order (<=8).
 Structure of finitely generated abelian groups. Free groups.
Applications.
References: Lang, Chapter I; Dummit and Foote, Part I; Rotman.
 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.
 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[x_{1},..., 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.
