Assume $f(x) \in \mathbb{Q}[x]$ is an irreducible cubic polynomial with discriminant $D$, let $x_1, x_2, x_3$ be the three distinct roots of $f(x)=0$. Let $p$ be any odd prime coprime to $D$. Are the following statements equivalent?
(1) $p$ is inert in $\mathbb{Q}[x]/(f(x))$.
(2) $p$ is inert in $\mathbb{Q}(x_1)$.
(3) $p$ is inert in $\mathbb{Q}(x_2)$.
(4) $p$ is inert in $\mathbb{Q}(x_3)$.
Of course, when $D$ is a square, these statements are equivalent. Here "inert" we mean $p$ still remains $p$ when factoring in the cubic extension.