Let $f(x)\in\mathbb{Q}[x]$ be an irrreducible cubic with root $\alpha$. Let $K=\mathbb{Q}(\alpha)$. There may be primes dividing $\text{disc}(f)$ that don't divide $\operatorname{disc}(K)$, so an intelligent choice of defining polynomial for $K$ is important. But even worse, $K$ may be non-monogenic, so that $\operatorname{disc}(K)$ is not equal to $\operatorname{disc}(g)$ for any cubic $g(x)$.
Is there a method to detect which primes divide $\operatorname{disc}(f)$ but not $\operatorname{disc}(K)$?
The motivation is that ramification of primes is most easily detected using a generating polynomial, but this may introduce extraneous primes i.e. primes that couldn't ramify since they don't divide $\text{disc}(K)$. This might lead to "false negative" answers to questions concerning ramification in $K$.
