Algebra I

Virtual Fall Semester 2020



Virtual Instructor: Matt Kerr
Empty Office: Cupples I, Room 114 [all office hours are on Zoom]
e-mail: matkerr [at] wustl.edu
Virtual Office Hours: Monday 8-9(PM), Wednesday 2:30-3:30, Friday 2-3

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
Topics include: integral domains, ideals, homomorphisms, fields, polynomial rings, Euclidean algorithm, multiplicative group of a finite field, principal ideal domains, unique factorization domains, Gauss's lemma, irreducibility tests, algebraic number rings.
IV. Modules
Topics include: Schur's Lemma, structure theorems for finitely generated modules over a PID and abelian groups, canonical forms, endomorphisms.
V. Algebras
Topics include: exterior algebras, division algebras, quaternions, Frobenius and Wedderburn theorems.

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 primarily groups, rings, and modules. In the Spring semester we turn to Galois theory, then go more deeply into the structure theory of rings and algebras, and apply this to representation theory and commutative algebra.

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

Class Schedule:

Lectures are on Tuesday and Thursday, from 1:00-2:20 PM, on Zoom. There will be some groupwork in breakout rooms since 80 straight minutes of online lecture is a bit much. First class is Tuesday Sept. 15 and last class is Thursday Dec. 17, with no class on Thursday Nov. 26 (Thanksgiving Holiday).

Midterm Exam: due Friday Nov. 13 (take home, covering groups and rings)
Final Exam: Thursday Jan. 7, 2021

The final exam is currently presumed to be a take-home exam as well, due on Jan. 7. If the virus situation changes and we are in person, it will take place on this date from 1-3 PM.

Assignments:

These will be due, via PDF upload to Canvas, on Tuesdays at noon. Solutions will also be posted on Canvas and may include students' work. I encourage you to visit my office hours to discuss problem sets, and to form study groups to discuss the more difficult problems (though solutions must be written up independently).

This is, of course, a "proof course". Please write your solutions legibly, linearly, logically and concisely, in complete sentences with proper mathematical grammar, and cite the results you use ("by Proposition (1.2.3) in the lecture notes/Jacobson, it follows that ..."). In addition, make sure to document any ideas that come from another text, person, or online source.

Problem Set 1 (due Tuesday Sept. 22)
Problem Set 2 (due Tuesday Sept. 29)
Problem Set 3 (due Tuesday Oct. 6)
Problem Set 4 (due Tuesday Oct. 13)
Problem Set 5 (due Tuesday Oct. 20)
Problem Set 6 (due Tuesday Oct. 27)
Problem Set 7 (due Tuesday Nov. 3)
Problem Set 8 (due Tuesday Nov. 10)
Problem Set 9 (due Wednesday Nov. 25)
Problem Set 10 (due Tuesday Dec. 1)

Grader: Xiaojiang Cheng


Lecture Notes:

Posted here as I write them, these will serve as our primary text, supplemented by the Jacobson book (see below).

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
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. Some results on finite groups
N. Burnside's counting 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 and polynomials over UFDs
L. Algebraic number rings
IV. Modules
A. Definition and examples


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.

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 Canvas.

Grades are typically curved in a course like this but will never be less than the following scale:

A+ A A- B+ B B- C+ C C- D F
TBA 90+ [85,90) [80,85) [75,80) [70,75) [65,70) [60,65) [55,60) [50,55) [0,50)

If you are a graduate student, a letter grade of B is required to pass; if you are an undergraduate taking this class Pass/Fail, you must earn a C- to pass.

The Washington University academic integrity policies are here. All work submitted under your name is expected to be your own; please make sure to document any ideas that come from another source.