Instructor: Tom Craven Phone: 956-4676
email: tom@math.hawaii.edu
Instructor's office hours:
Tuesday, Thursday 12:00-1:00, Friday TBA, and by appointment in Keller 307.
Class: Tuesday, Thursday 10:30-11:45 in Keller 413.
Class website: http://www.math.hawaii.edu/~tom/math611.html; you will
find a course syllabus and assignment list; later, grades will be posted here.
Texts: Algebra by Mark Steinberger, available online
here
Other useful references (standard texts):
Algebra by Lang
Algebra by Hungerford
Basic Algebra I and II by Jacobson
Algebra by MacLane and Birkhoff
We shall use the book as a reference and a source of examples and problems
rather than as a text to be followed closely. It has the advantage that
it covers the materially more thoroughly than most books, but is easier to
read than Lang or Hungerford.
The format for the course will be to have exercises due weekly. They will vary from routine applications of theorems covered in class to more challenging problems requiring more than a single sitting on your part to figure them out. Working on (struggling with) these exercises will be when most of your learning takes place. To encourage this struggle, your grade will be based mainly on the homework. There will also be a take-home final exam.
The following is a rough syllabus for the year. We may not do all of it.
Math 611:
Group theory (isomorphism theorems, Sylow theorems, composition series,
Jordan-Holder theorem, solvable groups, simplicity of An).
Language of category theory (object, morphism, universal object, functor,
coproducts, products, free objects).
Free groups.
Ring theory (definitions, UFD, PID, fraction field, polynomials).
Modules (products, coproducts and free modules, finitely generated modules
over a PID, noetherian and artinian modules, Jordan-Holder theorem).
Field theory (algebraic extensions, Galois theory, finite fields, cyclotomic
extensions, solvability, algebraic closure, transcendental extensions,
separability and inseparability).
Math 612:
Tensor products; projective, injective and flat modules; algebras; tensor
and exterior algebras.
Semisimple rings; Jacobson radical, simple and semisimple modules and
rings; Artin-Wedderburn theorem.
Projective modules, characterization of projective modules over artinian rings.
Commutative ring theory: nilradical, localization, prime spectrum.
Lattices, Boolean algebras, Boolean rings, Boolean spaces.
Bilinear forms, quadratic spaces and subspaces, isometries, orthogonal group,
Witt rings, quadratic forms over finite fields, ordered fields, Sturm's theorem,
Artin-Schreier theory.
Canonical forms for matrices
Computational algebra, Groebner bases.
Withdrawals from the class are allowed through Friday, October 24, 2008.