# Math 552 (Algebra), Spring 2022

### Prof. Weibel

Math 552 is the second semester of the standard introductory graduate algebra course. A significant portion of the course is devoted to the theory of field extensions, including Galois theory, rings of polynomials, Nullstellensatz, intro to homological algebra, structure of Artin rings. Additional topics are chosen in part based on the interests of the students.

• Instructor: Charles Weibel
• Text: Basic Algebra, by N.Jacobson.

• Lectures: TF2 (10:20 to 11:40PM) in Hill 425.

# Syllabus

The first 4 lectures are taught remotely.
• Galois Theory (Jan.18-Feb.22, 11 lectures)
• Midterm (Feb. 18)
• Commutative rings (March 1 to April 1)
• Midterm (April 1)
• Homological algebra, group representations
• Midterm (April 29)

# Homework Assignments (assigned weekly)

DueAssigned Problems
2/1Answer the 5 "why?" questions in Lecture 1 (on paper)
2/4Answer the 3+4=7 "why?" questions in Lectures 2,3 (Jan. 21,25)
2/11Answer the 1+3=4 "why?" questions in Lectures 4,5 (Jan. 28, Feb.1)
and do the Exercise in the Jan.28 notes
2/18 (Exam 1)From BAI: 4.8 #2,5; 4.9 #6; 4.15 #1,3,4,6; and 4.16 #8
2/22,25No class
3/8 (Tues)From BAII: 7.1 #2,3; 7.2 #1; 7.4 #1,2; 7.5 #1,3
3/22 (Tues)BAII: 3.7 #3,5; 7.6 #1,2; 7.7 #2
3/29 (Tues)BAII: 3.2 #1; 7.10 #2;
3) If Ri and R2 are noetherian, so is Ri×R2;
4) if R has ACC on {fin.gen. ideals}, show that R is noetherian;
5) If M is an R-module, and I=ann(x) is maximal among annihilators of nonzero elements of M, show that I is a prime ideal;
6) if R is noetherian and M a f.g. R-module, show that there is a chain of submodules Mi and primes Pi such that Mi/Mi+1≅R/Pi (hint: use 5)
4/1 (Fri) Midterm (Commutative rings)
4/12 (Tues) Group representations supplement, exercise (on p.2) and #1-3 (on p.4)
4/19 (Tues)Group representations supplement, on p.4: prove the Proposition and do exercises #4,5,6
4/26 (Tues) BAII: 3.10 #3,6; 6.2 #2; 6.7 #1; 6.8 #2    (problems 6.6 #1,2 are optional)
4/29 (Tues) Last class

An R-module P is projective if (P⊕Q) is a free R-module for some Q
An R-module P is flat if P⊗- is exact, i.e., when L is a submodule of M then L⊗P injects into M⊗P.

• Group representations supplement
• Bilinear forms supplement

Charles Weibel / Spring 2022