More specific. Consider the field extension $(L:K)$ and the inclusion of rings $\iota:K[x]\mapsto L[x]$. $Let f(x),g(x)\in K[x]$. Then, I have to proof this:
a)If $gcd(f,g)=1$ in $L[X]$, then $gcd(f,g)=1$ in $K[x]$
b)If $f(x)$ is irreducible in $L[x]$, then is irreducible in $K[x]$
Now, i have some hints for both exercises:
For (a), I have to suppose the opposite, i.e. $\exists\ q(x)\in K[X]$ s.t. $gcd(f,g)=q$ with $q\neq1$ nor unit (about $q$ being not unit I'm not so sure), but I don't see clearly what does this implies and how to proceed.
For (b), the hint is to use (a) in this case: I suppose $f$ is not irreducible in $K[x]$, then $\exists\ g,h\in K[x]$ s.t. $f=g\cdot h$, and the professor told us here that $gcd(f,h)\neq 1$ (here I use (a)'s contraposition), but I'm not sure how to proceed here neither.
In both cases I'm supposed to meet a contradiction somewhere in the process.