 
Summary: Problem Set 1
The next several problems are concerned with Sylvester's matrix and
the Sylvester determinant. Assume coecients are drawn from some xed
algebraically closed eld.
Problem 1. Let F1 = anxn
+ an1xn1
+ · · · + a1x + a0 and F2 = bmxm
+
bm1xm1
+ · · · + b0. Show that F1 and F2 share a root if and only if there
exists G, H with deg(G) = m  1, deg(H) = n  1 and GF1 = HF2.
Problem 2. Let S = {F1, xF1, . . . , xm1
F1, F2, xF2, . . . , xn1
F2} and let Pt
denote the vector space of all polynomials of degree less than or equal to t.
Use the result in Problem 1 to show that S is a linearly dependent set of
vectors in Pm+n1 if and only if F1 and F2 share a root.
Problem 3. By writing each element of S in terms of the basis {1, x, x2
, . . . , xm+n1
},
