Homework for Abstract Algebra II
Mathematics 567: Spring 2004


(Extremely) tentative syllabus. There is no doubt that we will veer from this at some point in the semester.

Week 1: 1/21-1/23, modules, vector spaces; 10.1, 11.1.
Week 2: 1/26-1/30, quotients, homomorphisms, matrices; 10.2, 11.2.
Week 3: 2/02-2/06, free modules, multilinear maps, dual vector spaces; 10.3, 11.4, 10.3.
Week 4: 2/09-2/13, tensor products; 10.4.
Week 5: 2/16-2/20, exact sequences, fund thm of fin gen modules over a PID; 10.5, 12.1.
Week 6: 2/23-2/27, canonical form; 12.2, 12.3.
Week 7: 3/01-3/05, field extensions; 9.4, 13.1, 13.2.
Week 8: 3/08-3/12, midterm on 3/8, independent time for projects 3/10 and 3/12.
Spring break
Week 9: 3/22-3/26, splitting fields, separable extensions; 13.4, 13.5.
Week 10: 3/29-4/02, fundamental theorem of Galois theory; 14.1, 14.2.
Week 11: 4/05-4/09, finite fields, cyclotomic extensions; 13.6, 14.3, 14.5.
Week 12: 4/12-4/16, simple and polynomial extensions; 14.4, 14.6.
Week 13: 4/19-4/23, insolvable and transcendental extensions; 14.7, 14.8, 14.9.
Week 14: 4/26-4/30, extra topics in Galois theory.
Week 15: 5/03-5/07, extra topics in Galois theory.

For every assignment, you may choose to hand in either the easy and medium problems or the medium and hard problems.

HW1: due Friday 1/30
10.1: easy 4; medium 10,11,19; hard 8, 23ab.
11.1: easy 1; medium 7,8.
11.2: easy 4; medium 35, 37ac; hard 12.

HW2
10.2: easy 1, 9; medium 4,5,6; hard 8.
10.3: medium 3,8,9,10; hard 7.
Extra problems from handout (medium) available here

HW3: due 2/13
10.3: easy 1; hard 15
11.3: easy 2cd; medium 2ab; hard 4,5.
11.2: easy 1,2; medium 8, 11.
Extra problems from handout (medium) available here

HW4: due 2/23 (Monday)
10.4: easy 2,13,27; medium 4,10,11,17,19,20; hard 5,15,18,24.
Extra problem from handout (medium) available here

HW4.5: due 2/27
Hand in a list of your two favorite topics for a project.

HW5: due 3/5
12.1: easy 13,16; medium 5,6,17,19; hard 9,14.
12.2: easy 7; medium 4, 9(ab), 14, 15; hard 5.
Extra problems from handout (medium) available here

Review Problems:
10.1: (Tor, Ann, submodule, F[x]-module) #8a, #9, #19; 10.2: (R-module homomorphism) #4, #9; 10.3 (cyclic, free, generate, direct sum) #3, HW2A; 10.4: (tensor) #11, #18.
11.1: (linear transformations) #8; 11.2 (matrices) #11; 11.3 (dual) #2ab.
12.1: (Big Thm, inv factors, elem divisors) #5, HW5 BC; 12.2 (Rational Canonical) #15; 12.3 (Jordan Canonical) #18.

HW5.5: due 3/21
Hand in a detailed outline for your project including: references, main results, main examples, and any questions you have.

HW6: due 3/26
12.3: easy 5; medium 18,22,31,36.
Extra problems from handout (medium) available here

HW7: due 4/2
9.4: easy 1cd, 5, 6ac; medium 2d, 8, 9, 13.
12.2: hard 20.
13.1: medium 3.
13.2: easy 1; medium 3,5,7,10,12,13.

HW8: due 4/9
13.4: easy 2; medium 4.
13.5: easy 2; medium 4,5.
13.6: medium 3,5,8a, 10,11, 12; hard 8bcd.

HW9: due 4/16
14.1: easy 1,3; medium 5,6,8; hard 9 (very challenging if char=p).
14.2: easy 1,3; medium 4,17,18,19; hard 5,21.

HW10: (last one!) due 4/28
14.3: easy 1; hard 7.
14.4: easy 2; medium 1,3.
14.5: easy 1; medium 3,7,10,12.
14.6: easy 23 (use 22); medium 46.
14.7: medium 4,5,6,10; hard 11.
14.9: medium 6.

Final Project: due 5/7.

Review Problems:
Chapters 10,11,12 see above.
9.4 #9.
13.1 #3, 13.2 #13, 13.4 #4, 13.5 #4, 13.6 #3, 10-12.
14.1 #1, 14.2 #4, 14.3 #1, 14.4, #3, 14.5 #3,10, 14.6 #46, 14.7 #10, 14.9 #6.