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Let $L$ be a number field and let ${\mathcal{O}}_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}_L$ to $k_B$ is surjective. What is the proof of this statement?

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By Ostrowski's theorem, the valuation in question must be the $P$-adic one for some prime $P$ of $\mathcal{O}_L$. This means that $k_B=\mathcal{O}_L/P$.

If you prefer to avoid using such a strong theorem, you can also use the characterization of valuation rings as maximal local subrings, since its maximal ideal can then be shown to be a prime of $\mathcal{O}_L$.

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