Lemma 10.119.10. Let $R \to S$ be a homomorphism of domains inducing an injection of fraction fields $K \subset L$. If $R$ is Noetherian local of dimension $1$ and $[L : K] < \infty $ then
each prime ideal $\mathfrak n_ i$ of $S$ lying over the maximal ideal $\mathfrak m$ of $R$ is maximal,
there are finitely many of these, and
$[\kappa (\mathfrak n_ i) : \kappa (\mathfrak m)] < \infty $ for each $i$.