Let $F, K$ be fields and suppose we have $F \subset K$ field extension. let $f(x) \in F[x]$ and $f'(x)$ its derivative. How can we prove that $gcd(f(x), f'(x)) = 1$ in $F$ $\iff gcd(f(x), f'(x)) = 1$ in $K$ ?
from right to left it's obvious because $F \subset K$, but I can't come up with a solution from left to right, nothing comes to my mind at all... Any help would be highly appreciated!
