Let F be a field, and let $p(x), q(x) \in F[x]$. If $p(x)$ and $q(x)$ have a common root c in some extension of F, they have a common factor of positive degree in $F[x]$
Here is my proof but I feel that I am not connecting all the dots:
Suppose p(x) and q(x) have a common root c in some extension of F. Let that extension field be E. We want to prove that p(x) and q(x) have a common factor of positive degree in F[x]. What it means for p(x) and q(x) to have a common root in c, p(x) = (x - c) a(x) and q(x) = (x - c) b(x). By the basic theorem of field extension, E contains all n roots of p(x) and q(x). From this though we have that p(x) and q(x) are elements of the kernel of the homomorphism of all polynomials p(x) and q(x) in F[x]. Thus p(c) and q(c) are in F(c). So a(c)|q(c) and a(c)|p(c) and b(c)|q(c) and b(c)|q(c). Thus a(c) = b(c) so there exists a minimal polynomial that is a factor of a(c) and b(c). By the fundamental homomorphism theorem the minimal polynomial which is a common factor is in F[x].
Any help would be great