Abstract Algebra
Mathematics 552 — Spring 2023

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.

Text: A Graduate Course in Algebra, by J. Brundan and A. Kleshchev.
Lectures: TF2 (10:20 to 11:40 AM) in Hill 423.

Syllabus
Galois Theory (Jan.17-Feb.17 (10 lectures)
Midterm (Feb. 21)
Commutative rings (Feb.24-Mar.10) Chapters 26-27 Spring Break (March 11-19)
Modules (Mar/21-31) Chapter 20
Midterm (March 31)
Homological algebra, group representations Chapters 19,22,23
Group representation supplement
Classes end May 1
Final exam May 2, 10:20-11:40AM

Homework Assignments (assigned weekly)

Due Assigned Problems
1/24 Answer the 5 "why?" questions in Lecture 1
1/31 Answer the 3+4=7 "why?" questions in Lecture 2,3
Find Gal(E/ℚ), where E is the splitting field of t-2.
2/7 Answer the 1+3=4 "why?" questions in Lectures 4,5 (Jan. 27, 31)
and do the Exercise in Lecture 4
2/14 Do the Exercises on pages 1 and 3 of Lecture 6
and the 3 Exercises in Lecture 7
2/21 1. Let f(t) be cubic with discriminant d, Galois group G. Show that G has 3 distinct real roots iff d<0.
2. Let E=ℚ(z), z⁵=1. Show that Gal(E/ℚ) is solvable, and find all subfields of E.
3. Let E=ℚ(√2,√5); find u so that E=ℚ(u).
4. If [E:F]=p and char(f)=p>0, show E=F(u), where u^p-u is in F.
2/28 Find Gal(E/ℚ), where E is the splitting field of t⁹-2.
Redo Exam1
3/7 From [BK]: 19.3.2, 19.3.16, 26.1.11, 26.2.14, 26.2.17
3/21 From [BK]: 26.2.30, 26.3.4, 26.3.15, 26.3.19, 26.4.15
3/28 From [BK]: 27.1.5, 27.2.7, 27.2.9, 27.2.15, 27.2.16, 27.2.21
4/7 From [BK]: 22.1.1, 22.1.3, 22.1.5
4/14 From [BK]: 22.2.6, 22.2.8; from supplement: Exercises 1,3,5
4/21
.
.
. From [BK]: 22.3.7, 22.3.8; (characters χ of groups)
Show that V=Fⁿ is a projective M_n-module;
Show that F(x) is an injective F[x]-module;
If I is the kernel of ℤ[G]→ℤ, show that I/I^2 ≅ G/[G,G].

Due	Assigned Problems
1/24	Answer the 5 "why?" questions in Lecture 1
1/31	Answer the 3+4=7 "why?" questions in Lecture 2,3 Find Gal(E/ℚ), where E is the splitting field of t-2.
2/7	Answer the 1+3=4 "why?" questions in Lectures 4,5 (Jan. 27, 31) and do the Exercise in Lecture 4
2/14	Do the Exercises on pages 1 and 3 of Lecture 6 and the 3 Exercises in Lecture 7
2/21	1. Let f(t) be cubic with discriminant d, Galois group G. Show that G has 3 distinct real roots iff d<0. 2. Let E=ℚ(z), z⁵=1. Show that Gal(E/ℚ) is solvable, and find all subfields of E. 3. Let E=ℚ(√2,√5); find u so that E=ℚ(u). 4. If [E:F]=p and char(f)=p>0, show E=F(u), where u^p-u is in F.
2/28	Find Gal(E/ℚ), where E is the splitting field of t⁹-2. Redo Exam1
3/7	From [BK]: 19.3.2, 19.3.16, 26.1.11, 26.2.14, 26.2.17
3/21	From [BK]: 26.2.30, 26.3.4, 26.3.15, 26.3.19, 26.4.15
3/28	From [BK]: 27.1.5, 27.2.7, 27.2.9, 27.2.15, 27.2.16, 27.2.21
4/7	From [BK]: 22.1.1, 22.1.3, 22.1.5
4/14	From [BK]: 22.2.6, 22.2.8; from supplement: Exercises 1,3,5
4/21 . . .	From [BK]: 22.3.7, 22.3.8; (characters χ of groups) Show that V=Fⁿ is a projective M_n-module; Show that F(x) is an injective F[x]-module; If I is the kernel of ℤ[G]→ℤ, show that I/I^2 ≅ G/[G,G].

Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9

Last modified 4/25/23

Charles Weibel / Spring 2023

Abstract Algebra Mathematics 552 — Spring 2023

Prof. Weibel

Syllabus

Homework Assignments (assigned weekly)

Last modified 4/25/23

Abstract Algebra
Mathematics 552 — Spring 2023