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.
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 

2/7  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), z^{5}=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 t^{9}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=F^{n} 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
Charles Weibel / Spring 2023