Math 427, Honors Abstract Algebra, Spring, 2006

My information:

Thomas Nevins
357 Altgeld Hall
217.265.6762
nevins@uiuc.edu
Office hours:  Tuesdays, noon to 1 p.m. in room 443 Altgeld; Fridays, 3:30-4:45 p.m. in my office (357 Altgeld); and by appointment

Syllabus (in PDF)
Midterm #1:  Friday, February 24 (tentative)
Midterm #2:  Friday, March 31 (tentative)

Final Exam:  Tuesday, May 9, 8--11 a.m.
Course Summary  This course will give an introduction to the basic objects of modern abstract algebra: groups, rings, and fields. We'll cover important structure theorems but will also focus our attention on a deep understanding of a few of the coolest examples in the subject.

A tentative list of topics:
Groups:  Definition, examples. Subgroups, cosets, normal subgroups and quotient groups, homomorphisms. Group actions, examples. Orbits, stabilizers under group actions. Isomorphism theorems. Cayley's theorem. Cauchy's theorem. Direct products.
Rings:  Definition, examples. Subrings, ideals, quotient rings, homomorphisms. Isomorphism theorems. Integral domains, fields, prime and maximal ideals. Geometric meaning of maximal ideals and quotient rings. Division algorithm, Euclidean domains. PIDs. Field of fractions of a domain. Modules, structure theorem for f.g. modules over a PID. Eisenstein criterion. Primes congruent to 1 mod 4 as sums of squares.
Some possible sources of examples: cyclic groups; (finite) p-groups; dihedral groups; symmetric groups, even and odd permutations, structure of permutations; matrix groups, especially SO(3) (and possibly GL(n,C)); finite subgroups of SU(2) (aka binary polyhedral groups). Polynomial rings; integers etc.; Gaussian integers; factor rings of polynomial rings and algebraic subsets of affine space; coordinate rings of plane conics and cubics. Rings of integers in some (quadratic) number fields.
A useful web page on groups of small order.
Homework:    [Guide to interpretation: T: "tractable,"  G: "of greater difficulty,"  V: "very challenging"]
Notes:
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