**Homework for Abstract Algebra**

Mathematics 566: Fall 2003

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

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.)

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.

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.

3.1, 4.1, 4.4, 4.5, 5.5, 7.3, 8.3.