# great common divisor

Prove: let $F$ be a field and let $g(x)$ and $h(x)$ be polynomials in $F[x]$ with $\text{gcd}(g(x),h(x))=1$. Let $a, b$ be elements of $F$ with $a\neq b$. Then $\text{gcd}(h(x)-ag(x), h(x)-bg(x))=1$.
Let $p(x)$ be an irreducible monic divisor of $h(x)-ag(x)$ and $h(x)-bg(x)$. Then $p(x)$ divides $bh(x)-abg(x)$ and $ah(x)-abg(x)$, and therefore also $[ah(x)-abg(x)]-[bh(x)-abg(x)]=(a-b)h(x)$ and $h(x)$. This in turn means $p(x)$ is a factor of $g(x)$. Since $0\leq\text{deg }p(x)\leq\text{deg gcd}(h(x),g(x))=0$, then $p(x)=1$, which implies the conclusion.