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$E$ is a field extension of $F$.
$p, q \in F[x]$ s.t. $p$ and $q$ are not relatively prime over $E$.
Show that $p$ and $q$ are not relatively prime over $F$.
Since $p$ and $q$ are not relatively prime over $E$, we have that $\gcd(p,q) \ne 1$ in $E[x]$.
Then let $d(x)$ be the greatest common divisor of $p,q$ in $E[x]$ which from (1) satisfies $\deg(d(x)) \ge 1$.
And from here I'm not sure how to proceed. Since $E$ is just a field extension over $F$, we don't even know if there exist roots of $d(x)$ in $E$, much less $F$. If necessary, we can extend $E$ to some field $K$ s.t. $K$ possesses all of the roots of $d(x), p(x)$ and $q(x)$ -- but I'm not sure how this helps us.