Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$? $K,L$ are fields, $K\subseteq L$. $f,g \in K[x]$. Suppose that $f,g$ are relatively prime as elements of $K[x]$. Prove they remain relatively prime  in $L[x]$.
I've tried everything I can think of. I feel like working with the contrapositive may be helpful but that's just a feeling.

Zev's answer is in some sense the canonical one, but here is another point of view, which is less elegant, but perhaps more intuitive.
We can embed $L$ into its algebraic closure $\overline{L}$; the algebraic closure
of $K$ in $\overline{L}$ is then an algebraic closure of $K$.
Now $f$ and $g$ coprime in $K[x]$ means that they have distinct roots in $\overline{K}$.  But these are also the roots of $f$ and $g$ in $\overline{L}$, and so $f$ and $g$ have distinct roots in $\overline{L}$.  Thus $f$ and $g$ are coprime in $L[x]$ as well.
