Math 552 (Algebra), Spring 2010
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:
Basic Algebra, by N.Jacobson.
Notices and assignments will appear on this page.  Lectures: MW4 (1:40 to 3:00PM) in Hill 423.
Instructor: Charles Weibel
Homework Assignments (assigned weekly)
Assigned  Problems  

4/26 (due 5/3) 
1. Let R be an infinite product of fields. Show that Spec(R)
is Hausdorff & totally disconnected. 2. If R⊂R' is an integral extension, Spec(R')→Spec(R) is a closed map. 3. Every noetherian valuation domain is a DVR. 4. BA II, 7.9 #3: R^{G} is finitely generated over F, when R is. (Here G is a finite group.)  
4/19 (due 4/26) 
1. BA II, 7.1 #4 (units in polynomial rings) 2. If h:R'→R and P is a prime ideal of R, show that h^{1}(P) is a prime ideal of R'. 3. Each prime ideal P of R is the kernel of a homomorphism R→F to a field F 4. If S is multiplicatively closed in R, there is a 11 correspondence between {ideals of S^{1}R} and {ideals I of R missing S with (sx∈I⇒x∈I) for all s∈x}.  
4/12 (due 4/19) 
1. Show that the tensor algebra satisfies
Hom_{Fmod}(V,A)≅ Hom_{Falg}(TV,A) 2. Let R be an infinite product of fields. Show that J(R)=0. Is R completely reducible as an Rmodule? 3. If E/F is a separable field extension, and D a division Falgebra, then E⊗_{F}D is semisimple.  
4/7 (due 4/12) 
1. BA II, 4.3 #4 (extending inclusions to endomorphisms) 2. If G is a finite pgroup and R=F_{p}[G], show that (a) the elements g1 are nilpotent, and they generate a nilpotent ideal of R; (b) there is only one simple Rmodule up to isomorphism. 3. For both groups of order 6 (D_{3} and C_{6}), give an isomorphism between the group ring Q[G] and a finite product of matrix rings over fields.  
3/31 (due 4/5) 
1. BA II, 3.5 #4 (completely reducible F[λ]modules). 2. BA II, 3.5 #7 (completely reducible modules of finite length). 3. For the following rings, (i) describe the simple Rmodules and (ii) prove that every Rmodule is completely irreducible: (a) a division ring D; (b) a matrix ring M_{n}(F) over a field F; (c) a finite product F_{1}×...×F_{n} of fields;  
3/24 (due 3/29) 
1. Show that the functor F(M)=M/IM is right exact on Rmod,
for any left ideal I of R
2. Let k be a field. Show that: (a) every left ideal I of M_{n}(k) has dim_{k}=r n for some r; (b) the left ideals of dimension n are in 11 correspondence with points of P^{n1}(k) 3. BA II, 3.1 #4 (short 5 lemma) 4. BA II, 3.2 #3 (artinian F{λ]module) 

3/10 (Wed)  Midterm on Galois Theory  
3/1 (due 3/8) 
1. Find the Galois group of the real field R over Q. 2. Jacobson 4.14 #1,4 (normal base for Q(√2,√3)) 3. Let E be the field Q(all √p, p prime). Find Gal(E/Q). Hint: Consider the subfields Q(√S) for various sets S of primes.  
2/24 (due 3/1) 
1. Jacobson 4.9 #7 (Casus irreducibilis) 2. Jacobson 4.11 #2 (discriminant of cyclotomic polynomial) 3. Jacobson 4.11 #3 (quadratic subfield of Q(ζ), p=1 or 3 mod 4) 4. Jacobson 4.6 #10: find the Galois group of x^{4}+3x^{3} 3x 2, using the fact that Gal(f/p) is a subgroup of Gal(f).  
2/17 (due 2/22) 
1. Show that x^{3} 3x+1 is either irreducible or has 3 roots
over any field; 2. Jacobson 4.8 #7 (the five possible Galois groups of a quartic); 3. Jacobson 4.6 #10 (Galois group of an explicit quartic);  
2/8 (due 2/15) 
1. Jacobson 4.6 #6 (nilpotent groups are solvable); 2. Jacobson 4.6 #10 (the two definitions of nilpotent agree); 3. Jacobson 4.6 #11 using #9 (nilpotent=product of pgroups)  
2/1 (due 2/8) 
1. Show that the Frobenius Φ has order n as an automorphism of
F_{q} when q=p^{n}. Conclude that Gal(F_{q}/F_{p}) is cyclic on Φ 2. Let E be the splitting field of x^{p}2 over Q. Show that Gal(E/Q) is the semidirect product (Z/p)⋊(Z/p)^{x} 3. Jacobson 4.5 #7. (find E^{G} if E=F_{p}(t) and t→t+1)  
1/25 (due 2/1) 
1. If F⊆E is a finite field extension,
any Fsubalgebra of E is a field. 2. Find a splitting field of x^{7}1 over F_{2} 

Last modified 04/28/10
Charles Weibel / Spring 2010