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
Homework Assignments (assigned weekly)
| Due | Assigned Problems | |
|---|---|---|
| 1/26 | Let E, E' be algebraic closures of a field F. Show any homom. E→E' fixing F is onto | |
| 2/2 | Show that Gal( F(x)/F ) ≅ PGL_2(F) (See BA I, Exercise 11 on p.243) | |
| 2/5 | Show that the Galois group of E/ Q is a semidirect product C7⋊ C6, where E is the splitting field of x7-2 | |
| 2/12 | For all primes p, determine Gal(E/Q), where E is obtained by adjoining all p-primary roots of 1 to Q. | |
| 2/16 | BA I 4.8 #5, 8(i,ii,iii). Galois groups of quartics. | |
| 2/23 | BA I 4.15 #1,6 (Norms and traces) | |
| 3/1 | BA II 8.7 #3,7 (about E⊗E and perfect fields) and: for E=Q(3√2) show that E⊗E = KxE as algebras. | |
| 3/4 | If f is irreducible, and E=frac(R), R=F[x0,...,xn]/(f)), show that tr.deg(E/F)=n | |
| 3/11 | First Midterm (Galois Theory) | |
| 3/29 | Show that center(Mn(D)=center(D); Show that H⊗H=M4(R); Find all subfields of H containing R. | |
| 4/1 | Group Representations Supplement, #3,4 (prove that the representations are irreducible) | |
| 4/8 | Compute W(Fq), q odd; Bilinear Forms Supplement, #2,3 | |
| 4/12 | BAII 7.1#2,4 and: If f:A→B, show that Spec(B)→Spec(A) is continuous. | |
| 4/15 | (BA II) Prove Prop 7.8 (p.401); 7.4#2; 7.6 #2 | |
| 4/19 | Show that Hom(-,P) is left exact; If B/A is integral, show that Spec(B)→Spec(A) is closed. | |
| 4/22 | Find the ring of integers in E=Q(√d), d odd squarefree. Prove that your answer is right. | |
| 4/26 | If A is a fin. gen. R-algebra and G=Gal(C/R), show that Max(A)=Max(A⊗C)/G (the orbit space). | |
| 4/29 | If L⊂M are torsionfree abelian groups, show that the kernel of L⊗Z/n→M⊗Z/n is isomorphic to Hom(Z/n,M/L) | |
Practice problems:
• Find the irreducible complex representations of G, the p-Sylow
subgroup of M3(Z/p), for p=2,3,5
• Find the ring of integers in E=Q(ξ), ξ5=1.
Hint: Find the discriminant of (ξ,...,ξ)5.
• If A is integrally closed in F, B is the integral closure of A in a
finite field extension of F and
P is a maximal ideal in A,
show that the ring B/PB is artinian.
• Given units a,b of a field F, let D be the vector space on basis
{1,x,y,xy} with multiplication
x^2=a,y^2=b,yx=-xy.
Show that D is a central simple F-algebra.
• Show that every finite abelian group A is (non-canonically)
isomorphic to Hom(A,Q/Z).
• If G is cyclic of order p2, show that
(Z/p[G])red=Z/p. Then show that
there is an
irreducible Z/p[G]-module M with MG≅(Z/p)2
and MG≅Z/p.
• Show that F2[C3] is a product of 2 fields.
What about F4[C3]?
• Find the Jacobson radical J of A=F2[S3],
and determine A/J. Hint: dim(J)=1.
Charles Weibel / Spring 2016