Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 20F / b / Solution

The equality of residue-field sizes says that the residue degree $f(Q/P)$ is one. Since $Q=\alpha O_L$, the principal ideal factorization of $\alpha$ has valuation one at $Q$ and zero at every other prime. Taking the relative ideal norm givesThus $P=N_{L/K}(\alpha )O_K$.