An outline for a proof of Theorem 17.8 follows.
(a) Let S = {s(x)f(x) + r(x)g(x)s(x), t(x) ∈ F[x]}. Select an element m(x) of minimum degree in S. (Recall that the zero polynomial has no degree, so it is not selected.) Can we guarantee that m(x) is monic?
(b) Show that if h(x) ∈ F[x] and h(x) divides both f(x) and g(v), then h(x) divides m(x).
(c) Show that m(x) divides f(x). If not, use the division algorithm and write f(x) = q(x)m(x) + r(x), where r(x) ≠ 0 and degree r(x) < degree m(x). Then show that r(x) ∈ S and obtain a contradiction.