Relatively Prime Polynomials in Extension Fields
The problem statement, all variables and given/known data
Recall that two polynomials f(x) and g(x) from F[x] are said to be relatively prime if there is no polynomial of positive degree in F[x] that divides both f(x) and g(x). Show that if f(x) and g(x) are relatively prime in F[x], they are relatively prime in K[x], where K is an extension of F.
The attempt at a solution
I'm guessing this will be a proof by contradiction where the contradiction will be that f(x) and g(x) are not relatively prime in F[x]:
Let p(x) be an irreducible factor of the divisor of f(x) and g(x) in K[x]. How can I show that p(x) divides f(x) and g(x) in F[x]?