$\gcd(f(x),g(x))\neq 1$ iff they share a root in some extension field

Question

Let $f(x),g(x)\in F[x]$ . Then to prove that $(f,g)\neq 1$ iff there is a field $E$ containing both $F$ and a common root of $f(x)\ \ and\ \ g(x)$

Now if $$(f(x),g(x))=d(x)\neq 1$$ then the splitting field field of $d(x)$ is our $E$ .

I am not sure about the converse . For if there exist an $E$ as said above then $f(x)$ and $g(x)$ have $\gcd$ say, $q(x)$ not equal to $1$ in $E$ . How do I pull back $q(x)$ in $F$ $?$ What will it look like in $F$ $?$ .

Hope I have conveyed my confusion properly .

Thanks for any help .