1. Let R = C[x_1..x_n], and R_d := the homogeneous polynomials of degree d.
For n=2, compute dim(R_d).
2. An ideal I in a polynomial ring is called graded if I is the sum over d of (I intersect R_d), i.e., if for every p in I, when we break p into its homogeneous components p_d each of those is also in I. Give an example of an ideal in C[z] that is not graded, and prove that it isn't.
3. If I,J are graded ideals, show I+J and (I intersect J) are graded too.
4. Let M_n(C) denote the noncommutative (!) ring of nxn complex matrices, where multiplication is matrix multiplication.
Fix T an element of M_n(C).
Define f : C[z] -> M_n(C), taking p(z) |-> p(T).
Let I = ker(f). This is principal (being an ideal of C[z]), i.e. I = < m(z) > for some m(z). What is this polynomial called in a linear algebra class?
Compute it for T = identity matrix.
5. Let p(z),q(z) be polynomials, and I = < p(z),q(z) >. Since I is principal, it's generated by some element r(z) = a(z)p(z) + b(z)q(z). What is this r(z) called? (some name involving p,q)
[Do you see how to use the division algorithm to compute the a(z),b(z)?]
6. If p,q are two polynomials in C[z] such that p^2 = q^3, show that there exists another polynomial r such that p = r^3, q = r^2. (Hint: use the Fundamental Theorem of Algebra, that every polynomial in C[z] factors as a number times a product of (z-a_i)^{m_i}, where the m_i is the multiplicity of the root a_i.)
7. Let R = C[x,y] / < x^2 - y^3 >. Use #6 to show that R is not isomorphic to C[z].
