If $\mathfrak{p}\subset \mathcal{O}_k$ is a prime ideal, and $M/L/k$ is a Galois extension tower such that $\mathfrak{p}|\mathcal{N}_{L/k}(c_{M/L}),$ where $c_{M/L}$ is the conductor of $M/L$. Is it true that if we know that $\mathfrak{p}$ splits completely in $L$ then all the primes that lie above $\mathfrak{p}$ divide the conductor, and why?
We may assume that $L/k$ and $M/L$ are abelian extensions, though $M/k$ need not be abelian.