## Relatively prime polynomials over a field have no common roots

Let $F$ be a field and let $p,q \in F[x]$ be relatively prime. Prove that $p$ and $q$ can have no roots in common.

We noted previously that $F[x]$ is a Bezout domain. In particular, if $p$ and $q$ are relatively prime then there exist $a,b \in F[x]$ such that $ap + bq = 1$. If $\zeta$ is a root of both $p$ and $q$, then we have $1 = 0$, a contradiction.