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.
Notices and assignments will appear on this page.

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)

Due Assigned Problems
2/1 Answer the 5 "why?" questions in Lecture 1 (on paper)
2/4 Answer the 3+4=7 "why?" questions in Lectures 2,3 (Jan. 21,25)
2/11 Answer 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
2/22,25 No 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 R_i and R₂ are noetherian, so is R_i×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 M_i and primes P_i such that M_i/M_i+1≅R/P_i (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

Due	Assigned Problems
2/1	Answer the 5 "why?" questions in Lecture 1 (on paper)
2/4	Answer the 3+4=7 "why?" questions in Lectures 2,3 (Jan. 21,25)
2/11	Answer 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
2/22,25	No 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 R_i and R₂ are noetherian, so is R_i×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 M_i and primes P_i such that M_i/M_i+1≅R/P_i (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