Mathematics 566: Fall 2011

Tentative syllabus
Week 1: C1, groups and examples of groups
Week 2: C1, group homomorphisms
Week 3: C2, labor day, subgroups
Week 4: C3, cosets, quotient groups
Week 5: C7, ring theory, in class work
Week 6: C3/C4, group actions
Week 7: C4, Sylow theorems
Week 8: C6, free groups, midterm
Week 9: C7, ring theory
Week 10: C7, ideals
Week 11: C8, principal ideal domains
Week 12: C9, in class work
Week 13: C8, unique factorization
Week 14: Special topics
Week 15: Special topics

Homework
HW1 (due 8/26) S0.1, S0.2, S0.3, S1.1,
HW2 (due 9/02) S1.2A, S1.3, S1.4, S1.5, S1.6
HW3 (due 9/09) S1.2B, S1.7, S2.1, S2.5
HW4 (due 9/16) S2.2, S2.3, S2.4
HW5 (due 9/23) S3.1, S3.2, S3.3
HW6 (due 9/30) S7.1, S7.2, S7.3A
HW7 (due 10/10) S3.4, S3.5, S4.1, S4.2, S4.3
midterm on 10/14
HW8 (due 10/28) S4.4, S4.5, S5.1, S5.2, S5.4, S5.5
HW 9 (due 11/4) S7.3, S7.4, S8.1, S8.2
HW 10 (due 11/14) S9.1, S9.2, S9.3
HW 11 (due 12/9) S8.3, 9.4, algebraic geometry

The problems to be handed in will be announced in class.

Chapter 0
S0.1: easy 2,4; medium 3,7.
S0.2: easy 5.
S0.3: easy 10; medium 11; hard 12,13,14.

Chapter 1
S1.1: easy 1(de),2(de),12,18; medium 6,8,9,20,22,29.
S1.4: easy 4; medium 8,10(abc); hard 7.
S1.5: easy 2(bc); medium 3.

S1.2A: easy 15; medium 2,3,7.
S1.3: easy 1,4b,9a; medium 12,16,18,20; hard 11.
S1.6: easy 7,8,11,15,21,22; medium 2,4,6,9,13,17,20; hard 25,26.

S1.2B: medium 9,10,12,13.
S1.7: easy 3,4a,6,11,15; medium 4b,5,8,13,16,17,18,20,21.

Chapter 2
S2.1: easy 1ace,2,3b,11; medium 1(bd),9,10a; hard 5*, 15.
S2.5: easy 9b.

S2.2: easy 2,3,12ab; medium 5ab,7,8,12cd; hard 12ef.
S2.3: easy 2,3,8,11,19; medium 9,12c,16,26.
S2.4: easy 5,6; medium 7,13 14cd; hard 10,16,18.

Chapter 3
S3.1: easy 3,12,17(a-c), 30,32, 37; medium 7,9,11bc,24,38,39; hard 14,40,41.
S3.2: easy 16, 23; medium 5,8,9,14,15; hard 4,18.
S3.3: easy 4; medium 1,8; hard 3.

Chapter 7
7.1: easy 11; medium 6, 24, 25, 27; hard 26, 30.
7.2: medium 3ab, 10, 12; hard 5b.
7.3A: easy 4, 7; medium 1, 6, 11, 12ab, 13.

Chapter 3 continued
S3.3: easy 4; medium 1; hard 8.
S3.4: medium 1; hard 2.
S3.5: easy 2; medium 7,13; hard 4.

Chapter 4
S4.1: medium 1,6,7; hard 8.
S4.2: easy 3; medium 4,5(a,c), 9; hard 7b,11.
S4.3: easy 3b, 10a; medium 5,8,11a; hard 22, 30.

Sample Midterm,

S1.1 22, S1.2 9, S1.3 20, S1.4 2, S1.5 1, S1.6 6, 13, 17, S1.7 16.
S2.1 9, S2.2 3, S2.3 9, 26, S2.4 14a.
S3.1 7, S3.2 18, S3.3 4, S3.4 1, S3.5 2.
S4.1 4, S4.2 4, S4.3 5, S4.4 3, S4.5 15.
S7.1 6, S7.2 12, S7.3 13

Chapter 4 continued
S4.4: medium 2, 3, 5.
S4.5: easy 4; medium 9,15,17, 40,45; hard 39, 46.

Chapter 5
S5.1: easy 1; medium 4 (up to generalize); hard 8,9.
S5.2: easy 1ab (list the groups), 10; medium 9, 14.
S5.4: medium 10; hard 14.
S5.5: easy 7ab; medium 8, 9; hard 6,11.

Chapter 7
S7.3B: medium 10,18,24.
7.4: easy 4; medium 9,14a-c, 15; hard 30,31a, 37, 38.

Chapter 8
8.1: easy 1b, 5a, 12; medium 7a, 8a (D=-2, D=-3 only).
8.2 medium 5.

Chapter 9
S9.1: medium 5,6,13.
S9.2: easy 8; medium 2,3,7.
S9.3: medium 3.

Last assignment
S8.3: medium 6ab, 8; hard 5.
S9.1: hard 11 (first part only).
S9.4: medium 1b, 6b, 7; hard 11.

Algebraic Geometry
Easy:
E1: Find the ideal corresponding to the union of the x and y axes in R^2.
Medium:
M1: If I is the ideal of C[x] generated by x^2(x-1)(x-3), find V(I) and rad(I).
M2: If I and J are ideals of C[x,y] and I is contained in J show that V(J) is contained in V(I).
M3: Consider the ring R[x,y] and its following ideals.
Let I_1=(f) where f=y-x^2. Let I_2=(g) where g=y-4+x^2. Let I_3=(fg) and let I_4=(f,g).
Draw V(I) for these four ideals.
Hard:
H1: If I is the ideal of C[x] generated by g(x) show I is radical if and only if the roots of g(x) are distinct.
H2: Find the ideal corresponding to the union of the x and y and z axes in R^3.
H3: Find the ideal corresponding to the union of the xy, yz and xz planes in R^3.

Sample Final,

Review Problems: Group review problems from above.
S5.1 5, S5.2, 9, S5.5 7a, S5.4 14, S5.5 11
S7.1 25c, S7.2 9c, S7.3 1, S7.4 2,9
S8.1 8a, S8.2 5, S8.3 6ab
S9.1 13, S9.2 3,6, S9.4 6b, S9.5 2a.

Hard Sections:
3.1, 4.1, 4.4, 4.5, 5.5, 7.3, 8.3.