GCD is same in a field and its superfield.

[caffeinemachine]

Let $K$ be an extension field of a field $F$ and let $p(t),q(t)\in F[t]$. Show that the monic greatest common divisors of $p(t)$ and $q(t)$ in $F[t]$ is same as the monic greatest common divisor of $p(t)$ and $q(t)$ in $K$.

 

[caffeinemachine]

Hint:

Let $p(t)$ and $q(t)$ have a non trivial common factor in $K[t]$. Assume that $p$ and $q$ don't have a non-trivial common factor in
$F[t]$. Then there exist $a,b\in F[t]$ such that $pa+qb=1$. But this contradicts the fact that $p$ and $q$ have a non-trivial common factor in $K[t]$.