if $\gcd(f,g)=1$ in $R[X]$, then $\gcd(f,g)=1$ in $K[X]$?
Let $R$ is an integral domain and $K$ its field of fractions
Question: if $\gcd(f,g)=1$ in $R[X]$, then $\gcd(f,g)=1$ in $K[X]$?
Note that $R$ is not necessary a GCD domain. $\gcd(f,g)=1$ means that the only common divisors of $f$ and $g$ are units of $R[X]$ (resp. $K[X]$).
Solution 1:
Try $R=\Bbb{Q}[t^2,t^3]$ and $t$ a root of both $f$ and $g$