Suppose $f$ is a nonzero polynomial over an arbitrary field. If $g=\gcd(f,f')$, is it true that $f/g$ is always separable?
I was trying to show $\gcd(f/g,(f/g)')=1$. If $d$ is a common divisor of $f/g$ and $(f/g)'$, then since $f=(f/g)\cdot g$, the product rule shows $$ f'=(f/g)'g+(f/g)g' $$ so $d\mid f'$ too. Since $d\mid f/g$, $d\mid f$, and finally $d\mid g$. On top of that, $d\mid f/g$ shows $dg\mid f$, and I was trying to conclude $dg\mid f'$ too to see that $dg\mid g$ to get that $d$ is a unit.