Algebra I

Fall Semester 2012



Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at] math.wustl.edu
Office Hours: Tues/Thurs 4-5:30, and by appointment

Course Outline:

I. Sets
A few preliminaries.
II. Groups
Topics include: subgroups, Cayley and Lagrange theorems, group actions and Burnside's lemma, orbits and conjugacy classes, cosets, normal subgroups, quotient groups, homomorphisms, Sylow theorems.
III. Rings and Modules
Topics include: integral domains, ideals, homomorphisms, polynomial rings, Euclidean algorithm, principal ideal domains, unique factorzation domains, Gauss's lemma, irreducibility tests, structure theorem, canonical forms, multiplicative group of a finite field, quaternions.
IV. Fields and Galois theory
Topics include: field extensions, splitting fields, normal and separable extensions, automorphisms and fixed fields, theorem of the primitive element, solvability and composition series, Galois groups, applications.

This is the first half of a year-long course which forms the basis for the Ph.D. qualifying examination in algebra. This semester we will cover much of Chapters 1-4 of the classic book by Jacobson. In the Spring semester we will finish Galois theory, then go more deeply into the structure theory of rings and algebras, and apply this to representation theory, commutative algebra, and homological algebra.

Prerequisites: Math 430 or the equivalent, or permission of the instructor.

Class Schedule:

Lectures are on Monday, Wednesday and Friday, from 2-3 PM, in Cupples I Rm. 207. First class is Wednesday Aug. 29 and last class is Friday Dec. 7, with holidays on Sept. 3 (Labor Day), Oct. 19 (Fall Break), and Nov. 21-23 (Thanksgiving Holiday).

Midterm Exam: Friday Oct. 26 (in class) (solutions)
Final Exam: Monday Dec. 17, 3:30-5:30 (solutions)

Both exams are in Rm. 207. The midterm will cover approximately Chap. 1-2 in Jacobson.

Assignments:

These will be collected on Fridays and returned the following Monday or Wednesday. Solutions will also be posted and will include students' work. Please feel free to come to office hours to discuss problem sets. Make sure you can do any problems I don't assign.

Problem Set 1: hand in 5-10 (due Friday Sept. 7) (solutions)
Problem Set 2: hand in all (due Friday Sept. 14) (solutions)
Problem Set 3: hand in all (due Friday Sept. 21) (solutions)
Problem Set 4: hand in all (due Friday Sept. 28) (solutions)
Problem Set 5: hand in all (due Friday Oct. 5) (solutions)
Problem Set 6: hand in all (due Friday Oct. 12) (solutions)
Problem Set 7: hand in all (due Friday Oct. 19) (solutions)
Problem Set 8: hand in all (due Friday Nov. 2) (solutions)
Problem Set 9: hand in all (due Friday Nov. 9) (solutions)
Problem Set 10: hand in all (due Friday Nov. 16) (solutions)
Problem Set 11: hand in all (due Friday Nov. 30)
Problem Set 12: hand in all (due Friday Dec. 7)

Grader: Ryan Keast


Lecture Notes:

Will be scanned and posted here as I write them. The hope is that this makes taking notes optional.

I. Sets
A. Relations
B. Integers
C. Posets
II. Groups
A. Introduction
B. Permutation groups
C. Groups and Subgroups
D. Cosets and Lagrange's Theorem
E. Homomorphisms and Isomorphisms
F. Group actions and Cayley's theorem
G. Conjugacy and the orbit-stabilizer theorem
H. Cauchy's theorem
I. Normal subgroups and quotient groups
J. Automorphisms
K. Generators and Relations
L. The Sylow Theorems
M. Results on finite groups
N. Not-Burnside's Lemma
III. Rings
A. Examples of Rings
B. Ring Zoology
C. Matrix Rings
D. Ideals
E. Homomorphisms of Rings
F. Fields
G. Polynomial Rings
H. Principal Ideal Domains
I. Unique Factorization Domains
J. Greatest Common Divisors
K. Gauss's Lemma
L. Algebraic Number Rings
IV. Modules
A. Definition and Examples
B. Submodules and Homomorphisms
C. Modules over a PID
D. Applications to Linear Algebra
E. Endomorphisms
V. Remarks on Associative Algebras
A. Algebras over a Field
B. Finite-dimensional Division Algberas


Books:

Nathan Jacobson, Basic Algebra I (2nd Ed.); Dover

is the recommended textbook, which means I will follow it at least half the time and some of the problems I assign will come from it. Buy a copy of this (and Basic Algebra II for next semester) -- it's cheap.

If you would like to read more adventurously than Jacobson and/or my lecture notes, here are some suggestions. First, there are many other excellent standard texts, including

D.J.H. Garling, A Course in Galois Theory; Cambridge Univ. Press,

Thomas Hungerford, Algebra; Springer-Verlag,

Serge Lang, Algebra; Springer-Verlag,

and

Joseph Rotman, Advanced Modern Algebra; Prentice-Hall.

I will place one copy of each of these books on reserve at the Olin Library.

Grading Policy:

Your final grade for the semester is determined as follows: HW 40%, midterm 20%, final exam 40%. I will drop the lowest two grades you receive on homework.

Homework and examination grades will be regularly updated on telesis.