Homework for Abstract Algebra
Mathematics 566: Fall 2003

HW1 (due 8/29) S0.1, S0.2, S0.3, S1.1, S1.4, S1.5
HW2 (due 9/05) S1.2, S1.3, S1.6,
HW3 (due 9/12) S2.1, S1.7, S2.5, plus more problems from S1.2
HW4 (due 9/19) S2.3, S2.4, S2.2
HW5 (due 9/26) S3.1, S3.2
HW6 (due 10/6) S3.3, S3.4, S3.5, S4.1
HW7 (due 10/13) S4.2, S4.3, S4.4
HW8 (due 10/22) S4.5, S5.1, S5.2
HW9 (due 10/27) S5.4, S5.5, S6.3
HW10 (due 11/7) S7.1, S7.2
HW11 (due 11/14) S7.3, S7.4
HW12 (due 11/21) S8.1, S9.1
HW13 (due 12/5) S8.2, S8.3, S9.2
HW14 (due 12/12) mostly 9.4 plus random.

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

HW1
S0.1: easy 2,3,4; medium 7.
S0.2: easy 5.
S0.3: medium 10,11; hard 12,13,14.
S1.1: easy 1(de), 2(de), 11, 12; medium 5,6,9,18,20,22.
S1.4: easy 1,2,3; medium 4,8,10(abc); hard 7.
S1.5: easy 2(bc).

HW2
S1.2: medium 2,3,7.
S1.3: easy 1,6; medium 7, 9a, 18, 20; hard 11, 17.
S1.6: easy 7,11; medium 2,6,13,17; hard 9, 26.

HW3
S1.2: medium 9,10,11; hard 12.
S1.7: easy 1,3,11; medium 4a,5,6,8,16,17,20; hard 9,10,21.
S2.1: easy 1a,2a,3b; medium 1(bd) 9,10a; hard 5*, 15.
S2.5: easy 9a.

HW4
S2.3: easy 3,8,11; medium 9,12c,16,19,26; hard 23.
S2.4: easy 6; medium 3,7,12, 14cd; hard 10, 16a.
S2.2: easy 4, 12a; medium 3,7,8, 12bc; hard 12d-f.

HW5
S3.1: easy 3,12,17(a-c), 37; medium 7,9,11,24,30,39,40; hard 14,38,41.
S3.2: easy 16, 23; medium 5,8,9,14,18; hard 4,11.

HW6
S3.3: read carefully, easy 4; medium 8; hard 3.
S3.4: easy 2; medium 1, 4; hard 9.
S3.5: easy 13; medium 2,4,9,10; hard 7.
S4.1: medium 1,4,5b, 6.

HW7
S4.2: easy 3; medium 4,5(a,c), 9; hard 7b,11.
S4.3: easy 2c, 3b, 10a; medium 5,6,11a, 13; hard 8, 30.
S4.4: easy 3; medium 2, 13.

HW8
S4.5: easy 4; medium 9,15,17; hard 11.
S5.1: medium 1,4 (up to generalize), 5.
S5.2: easy 1ab (list the groups); medium 9,10; hard 14.

Review Problems
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.
S5.1 5, S5.2, 9, S5.5 7a.

HW9
S5.4: medium 10,14.
S5.5: easy 7a; medium 1, 7b, 11; hard 9.
S6.3: medium 2,4.

HW10
S7.1: easy 11; medium 6, 24, 25, 30; hard 26bc, 27.
S7.2: easy 1c; medium 3ab, 10, 12; hard 5b.

HW11
S7.3: easy 6; medium 1, 4, 10, 17, 18a, 34ab; hard 11, 34c.
S7.4: easy 4, 16; medium 2, 9, 14a-c, 15; hard 30, 31a, 38.

HW12
S7.4: hard 37, 38.
S8.1: easy 1b,5a; medium 7,8a (just for D=-3),12; hard 9.
S9.1: medium 5,6,13.

HW13:
S8.2: medium 3,4,5.
S8.3: medium 6ab, 8; hard 5.
S9.2: medium 2,3,6,8; hard 7.

HW14:
S9.1: medium 4; hard 11 (first part only).
S9.3: medium 3.
S9.4: easy 14; medium 1ab, 6b, 7; hard 11.
S9.5 medium 2a.

Extra credit
Elliptic Curves
1. Define a composition law on the cubic curve E by the rule P*Q is the third point of intersection of E with the line through P and Q.
Prove that E is NOT a group under this composition law.
2. Let E be the cubic curve y^2=x^3-43x+166 and let P=(3,8). Compute 2P, 4P, and 8P. By comparing 8P with P, find the order of P.
3. If the identity is chosen to be the point at infinity, where are the points of order 2? How many are there?
(The answer should depend on f(x) where y^2=f(x) is the cubic curve.)

Algebraic Geometry
1. easy: If I is the ideal of C[x] generated by x^2(x-1)(x-3), find V(I) and rad(I).
1. hard: 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.
2. easy: 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).
2. hard: Prove V(I) union V(J) equals V(IJ).
3. medium: 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.
4. easy: Find the ideal corresponding to the union of the x and y axes in R^2.
4. hard: Find the ideal corresponding to the union of the x and y and z axes in R^3.
4. hard: Find the ideal corresponding to the union of the xy, yz and xz planes in R^3.

Review Problems: Group review problems from above.
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.