Algebra II

Virtual Spring Semester 2021



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: 8-9pm M, 2:45-3:30 W, 4-4:45 F

Course Outline:

I. Galois Theory
II. Linear Algebraic Groups
III. Representation Theory
IV. Commutative Rings

This is the second half of a year-long course which forms the basis for the Ph.D. qualifying examination in algebra.

Prerequisites: Math 5031, or permission of the instructor.

Class Schedule:

Lectures are on Monday, Wednesday and Friday, from 2-2:45 PM, on Zoom. First class is Monday Jan. 25 and last class is Monday May 3. There is no class on Mar. 3, Mar. 22, or April 12 (study/wellness days).

Midterm Exam: due Friday Mar. 19 (take home)
Final Exam: Tuesday, May 11

The final exam is currently presumed to be a take-home exam as well, due on May 11. If the virus situation changes and we are in person, it will take place on this date from 3:30-5:30 PM.

Assignments:

These will be due, via PDF upload to Canvas, on Tuesdays at 5pm. 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 Feb. 2)
Problem Set 2 (due Tuesday Feb. 9)
Problem Set 3 (due Tuesday Feb. 16)
Problem Set 4 (due Tuesday Feb. 23)
No HW due week of March 1 (wellness days)
Problem Set 5 (due Tuesday Mar. 9)
Problem Set 6 (due Tuesday Mar. 16)
Problem Set 7 (due Tuesday Mar. 30)
Problem Set 8 (due Tuesday Apr. 6)
Problem Set 9 (due Wednesday Apr. 14)
Problem Set 10 (due Tuesday Apr. 20)
Problem Set 11 (due Tuesday Apr. 27)
Problem Set 12 (due Tuesday May 4)

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. Galois Theory
A. Field extensions
B. Constructible points
C. Splitting fields
D. Algebraic closures
E. Multiple roots
F. Separable, normal, and Galois extensions
G. Automorphisms and fixed fields
H. Finite fields
I. Simple extensions
J. Solvable groups and radical extensions
K. Discriminants, cubics, and quartics
L. Higher degree
M. Norms and traces
N. Transcendental extensions
II. Linear algebraic groups
A. Bilinear forms
B. Zoology of classical groups
C. Symplectic groups
D. Quadratic forms and orthogonal groups
E. Hermitian forms and unitary groups
III. Representation theory
A. Semisimple modules and rings
B. Artin-Wedderburn theorem
C. Semisimple algebras
D. Characters of representations
IV. Commutative rings
A. Localization
B. Chain conditions
C. Primes and radicals
D. Primary decomposition
E. Nakayama's lemma
F. Ring extensions
G. Affine varieties
Link to Algebra I course notes.

Books:

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

is the recommended textbook. We will in large part follow it and use exercises from it for the treatment of Galois theory and bilinear forms (but not for the later parts of the course).

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.