Algebra II

Spring Semester 2025


Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at] wustl.edu
Office Hours: 12-12:50 M, 3-3:50 W
Course Description:

Continuation of Math 5031. Prereq: Math 5031 or permission of instructor.
Course Outline/Learning Goals:

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

I. Galois Theory
Topics include: splitting fields, Galois extensions/groups/correspondence, geometric applications, solvability, primitive element theorem, reduction mod p and Frobenius map, Lindemann-Weierstrass (Jan. 13 - Mar. 3)
II. Linear algebraic groups
Topics include: symmetric/alternating/Hermitian forms, Sylvester's theorem, orthogonal/symplectic/unitary groups, transvections and reflections (Mar. 5 - Mar. 24)
III. Representation Theory
Topics include: semisimplicity, Artin-Wedderburn, Maschke's theorem, central simple algebras, character theory, group cohomology (Mar. 26 - Apr. 4)
IV. Commutative Rings
Topics include: localization, Noetherian and Artinian rings, Nakayama's lemma, Hilbert basis theorem, integral closure, Nullstellensatz (Apr. 7 - Apr. 25)
Class Schedule:

Lectures are on Monday, Wednesday and Friday, from 11-11:50 AM, in Cupples II Room 200. First class is Monday Jan. 13 and last class is Friday Apr. 25; MLK holiday is Jan. 20, and Spring Break is March 9-16.

Midterm Exam: Wed. March 19, 11AM-11:50AM (in class, covering Galois Theory)
Final Exam: May 6, 2025, 10:30AM-12:30PM (in the same classroom, cumulative)
Assignments:

These will be due by Gradescope submission Thursday by 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 Thursday Jan. 23)
Problem Set 2 (due Thursday Jan. 30)
Problem Set 3 (due Thursday Feb. 6)
Problem Set 4 (due Thursday Feb. 13)

Grader: Calvin Reedy, c.m.reedy [at] wustl.edu
Lecture Notes:

I will follow the notes below, which will serve as our primary text, supplemented by the Jacobson book (see below). Here is a link to the Algebra I notes for easy reference.

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
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.
Additional Resources:

See the attached Spring 2025 University Policies and Resources document.
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.