I am solving exercise 3.37 (e) in Number Fields by Marcus. Let $S/R$ be an extension of Dedekind domains, $P$ is a prime of $R$ and $Q$ is a prime of $S$ lying over $P$. Let $e=e(Q|P)$, then in (d) I showed that $Q^{e-1}~|$ diff $S$. Now in (e) I have to show that $Q^{e-1}~|~f'(\alpha)S$ for any $\alpha\in S$, where $f$ is the monic irreducible polynomial for $\alpha$ over $R$.
I think that using (d) it's sufficient to show that $f'(\alpha)\in$ diff $S$. Since diff $S=(S^*)^{-1}$, it turned to show that $f'(\alpha)s\in S$ for all $s\in S^*$. However I have no idea for this. I don't know how can I deal with $f'(\alpha)$. Can anyone please show me some hints? Thank you in advance!
