Math 251: Groups, rings and fields
Instructor: C. Schoen

This is an introductory algebra course intended primarily for first year
graduate students. It provides indispensable background material for
graduate level courses and research in algebra, algebraic geometry,
algebraic topology, commutative algebra, number theory, and some parts of
complex analysis. There are also applications to differential geometry and
analysis. Students should have taken an undergraduate algebra course which
covered equivalence relations and introduced abstract groups. A good command
of quotient constructions (e.g. for vector spaces) would be very helpful.
The course will begin with commutative rings, discuss factorization in such
rings, and then proceed to the study of modules over such rings. The theory
of finitely generated modules over principal ideal domains will be treated
thoroughly. The theory of field extensions will be introduced. Finite fields
will be classified. Galois theory will be developed carefully. Group theory
will be reviewed quickly as needed for applications to Galois theory. The
course will roughly follow the five of the last six chapters of the text
Algebra, by Michael Artin (second edition). A substantial written assignment
with most problems coming from Artin's book may be expected each week.