Let $K$ be an algebraic number field. Let $A$ be the ring of integers in $K$. Let $L$ be a cyclic extension of $K$ of degree $l$, where $l$ is a prime number. Let $B$ be the ring of integers in $L$. Let $G$ be the Galois group of $L/K$. Let $\sigma$ be a generator of $G$. Let $\mathfrak{D}_{L/K}$ be the relative different of L/K.
My question: Is the following proposition true? If yes, how would you prove this?
Proposition Let $\mathfrak{P}$ be a prime ideal of $B$. Let $\mathfrak{p} = \mathfrak{P} \cap A$. Then $\mathfrak{P}$ divides $\mathfrak{D}_{L/K}$ if and only if $\sigma(\mathfrak{P}) = \mathfrak{P}$ and $\mathfrak{p}B \neq \mathfrak{P}$.
Related question: Selfconjugate prime ideal of a cyclic extension of an algebraic number field of prime degree.