MATH 439/739/819 Assignment 5. Spring 2010. Due Monday March 29th at 10:30am. Late policy: -20% for up to 24 hours late. Zero for more than 24 hours late. Problems marked * are for MATH 819 students only. Michael Monagan Problems from the text. 4.3 # 6(a), 8, 9 4.4 # 1, 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 do this. Theorem 6 of section 4.6 says that there exist polynomials f and g such that I: = K and I: = J. Find such f and g. Additional exercises 1: Read the material on ``Products of Ideals'' on pages 185-186. Notice that the product I.J of two ideals I and J is NOT defined to be the SET of products S = { f g : f in I and g in J }. Find an example which illustrates that S is not closed under addition. 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(q) is the finite field with q 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) in R[x], (ii) in R[x], (iii) in R[x,y] (iv) in R[x,y], and (v) 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 = 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 U and V. 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 = . Decompose I into a minimal intersection of prime ideals. 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* By computing Groebner bases and factoring polynomials, compute the radical of the ideal 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 >