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?



The map from the ring of integers to the residue field of a valuation subring is surjective
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 $\mathcal{O}\_L$.


