Prime below prime is unique

Question

Let $L/K$ be an extension of algebraic number fields, and let $\mathcal{O}_L$ and $\mathcal{O}_K$ be the corresponding rings of integers. Then $\mathcal{O}_K$ is a subring of $\mathcal{O}_L$, and for any prime ideal $\mathfrak{p} \subset \mathcal{O}_K$, there exists a (necessarily unique) factorisation $$ {\mathfrak p}{\mathcal O}_L= \mathfrak{P}_i^{e_i}\cdots\mathfrak{P}_r^{e_r} $$ into prime ideals $\mathfrak{P}_i \subset \mathcal{O}_L$.

The prime ideal $\mathfrak{P}_i$ in the factorisation of $\mathfrak{p}$ is said to be unramified over $\mathcal{O}_K$ if $e_i = 1$; i.e. it appears only once in the factorisation of $\mathfrak{p}$.

But then one speaks of primes $\mathfrak{P} \subset \mathcal{O}_L$ being unramified (or ramified) over $\mathcal{O}_K$ without any reference to the prime in $\mathcal{O}_K$ below them.

This would seem to suggest that for any prime $\mathfrak{P} \subset \mathcal{O}_L$, the prime ideal in $\mathcal{O}_K$ below it is unique.

Is that the case? I cannot seem to find anything on it.

Does anyone know where I could find a reference?

Answer 1

Good observation, Heinrich. Indeed it is true that there is a unique prime ideal of ${\mathfrak p}\subset{\mathcal O}_K$ below any prime ideal ${\mathfrak P}\subset {\mathcal O}_L$: It's simply ${\mathfrak p} := {\mathfrak P}\cap {\mathcal O}_K$. This follows from two facts:

1. For any ${\mathfrak p}\subset{\mathcal O}_K$, the $L$-primes ${\mathfrak P}$ appearing in the decomposition of ${\mathfrak p}$ are precisely those for which ${\mathfrak p}{\mathcal O}_L\subset {\mathfrak P}$ (in ${\mathcal O}_L$), or equivalently ${\mathfrak p}\subset {\mathfrak P}\cap{\mathcal O}_K$ (in ${\mathcal O}_K$).

2. For any prime ${\mathfrak P}\subset{\mathcal O}_L$, ${\mathfrak P}\cap {\mathcal O}_K$ is prime and non-zero.

Answer 2

You can use the fact that a Dedekind Domain has Krull dimension one, that is, any non zero prime ideal is maximal. Let $\mathfrak{P}$ be a non zero prime ideal of $\mathfrak{O}_L$. Then $\mathfrak{p} := \mathfrak{P} \cap \mathfrak{O}_K$ is a non zero prime in $\mathfrak{O}_K$, hence maximal and thus unique.