I have statements like unramified extensions of number fields are examples of etale morphisms. How exactly do unramified extensions of number fields give rise to etale morphisms?
Etale morphisms and unramified extensions of number felds
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algebraic-geometry
algebraic-number-theory
etale-cohomology
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1What definition of etale morphism do you know? When you know that etale means flat and unramified it should be easy. – 2017-02-18
1 Answers
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Hint: Note that if $L/K$ is an extension of number fields and $\mathfrak{p}$ is a prime of $K$, then $\mathfrak{p}(\mathcal{O}_K)_\mathfrak{q}=\mathfrak{q}^{e(\mathfrak{q}\mid\mathfrak{p})}$ for every prime $\mathfrak{q}\mid\mathfrak{p}$. Recall then that what if means for a map $\text{Spec}(\mathcal{O}_L)\to\text{Spec}(\mathcal{O}_K)$ to be unramified at $\mathfrak{q}$ is that if $\mathfrak{p}$ is its image, then $\mathfrak{p}\mathcal{O}_{\text{Spec}(\mathcal{O}_L),\mathfrak{q}}$ is the maximal ideal of $\mathcal{O}_{\text{Spec}(\mathcal{O}_L),\mathfrak{q}}$. Compare that to my first sentence and see what you get.