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 $i$.

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