MATH 498/814 Assignment 5.  Spring 2001.
Due Friday July 21st at the beginning of class.
Late policy: -10% for each day late.
Problems marked * are for MATH 800 students only.
For this assignment, you MUST use Maple when appropriate.  Also, you may only
use commands in the PolynomialIdeals package to check your answers.
Michael Monagan

Problems from the text.
4.3 # 6(a), 8, 9
4.4 # 1(a), 2, 3, 6
4.5 # 2, 10*
4.6 # 2, 4, 7

Notes on exercises

For problem #6 of 4.4 it is sufficient to show that (2) and (3) hold for r=2.
For problem #7 of 4.6 you are to decompose I into the intersection of,
it turns out, two prime ideals J and K.  You must verify that J and K are
prime ideals and also that I:J = K and I:K = J.
Use the Quotient command in the PolynomialIdeals package to help do this.
Theorem 6 of section 4.6 says that there exist polynomials f and g
such that I:<f> = K and I:<g> = J.  Find such f and g.

Additional exercises

1: Proposition 2 of section 4.3 states that I+J is the smallest ideal
   containing I and J.  Rewrite the proof using ``proof by contradiction''.

2: For problem #12 of section 4.2 you computed g in Q[x,y], a GCD of three
   polynomials, namely, f, diff(f,x), and diff(f,y) using Maple's gcd command.
   Apply Theorem 11 and Proposition 13 of section 4.3 to compute g.

3: Let f,g be in GF(p)[x1,...,xn] where GF(p) is the finite field on
   p elements.  Let S = V(f) - V(g).  Is S an affine variety?
   If so give a reason.  If not give a counter example.

4: Which of the following ideals are prime and which are maximal?
   (i) <x^3+1> in R[x],  (ii) <x^4+1> in R[x], (iii) <x^2+1> in R[x,y]
   (iv) <x^2+1,y^2+1> in R[x,y], (v) <x^2+1,y^2> in R[x,y]

5: Let C be the set of complex numbers and R be the set of real numbers.
   (i) Identify which ideals in C[x] are maximal?   No proof needed.
   (ii) Identify which ideals in R[x] are maximal?   No proof needed.

6: Let I = <y*x-x^3, z-x^3> and let W be the variety of I in C^3.
   This variety W is the union of two irreducible varieties U and V.
   Our problem is to determine what the two varieties are.
   Determine J = I(U) and K = I(V) either by inspection or by
   computing an appropriate Groebner basis G for I in Maple and
   factoring the polynomials in G if you need to.
   Now compute I:K and compute I:J and verify that J = I:K and K = I:J .

7: Consider the following ideal I = <x^2-y,y^4-y*z^2,x*y^3-x*z^2>.
   Decompose I into an intersection of prime ideals which is minimal.
   You should get three prime divisors of I.
   To find the decomposition use Maple to factor the polynomials that
   appear in Groebner bases for I.  Use the ideal quotient operation 
   to help you identify prime ideals which divide I.
   Verify that the decomposition of I is minimal (irredundant).

8: (bonus question).
   Compute the radical of the ideal generated by
   I = < (z^2-2)*(z^2-3),  y^5+y^3*z+y^3-3*y^4*z-3*y^2*z^2-3*y^2*z+
              3*y^3*z^2+3*y*z^3+3*y*z^2-z^3*y^2-z^3+6-5*z^2 >