# 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)

DueAssigned Problems
1/24Answer the 5 "why?" questions in Lecture 1
1/31Answer the 3+4=7 "why?" questions in Lecture 2,3
Find Gal(E/ℚ), where E is the splitting field of t-2.
2/7Answer the 1+3=4 "why?" questions in Lectures 4,5 (Jan. 27, 31)
and do the Exercise in Lecture 4
2/14Do the Exercises on pages 1 and 3 of Lecture 6
and the 3 Exercises in Lecture 7
2/211. 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 up-u is in F.
2/28Find Gal(E/ℚ), where E is the splitting field of t9-2.
Redo Exam1
3/7From [BK]: 19.3.2, 19.3.16, 26.1.11, 26.2.14, 26.2.17
3/21From [BK]: 26.2.30, 26.3.4, 26.3.15, 26.3.19, 26.4.15
3/28From [BK]: 27.1.5, 27.2.7, 27.2.9, 27.2.15, 27.2.16, 27.2.21
4/7From [BK]: 22.1.1, 22.1.3, 22.1.5
4/14From [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 Mn-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].