Prove: let F be a field and let g(x) and h(x) be polynomials in F[x] with gcd(g(x),h(x))=1. Let a, b be elements of F with a‡b. Then gcd(h(x)-ag(x), h(x)-bg(x))=1.
Prove: let be a field and let and be polynomials in with . Let be elements of with . Then .
Let be an irreducible monic divisor of and . Then divides and , and therefore also and . This in turn means is a factor of . Since , then , which implies the conclusion.