Let $o$ be a Dedekind domain, let $K:=Quot(o)$ denote its quotient field, let $L/K$ be a finite separable field extension, let $O\supset o$ be the integral closure of $o$ in $L$. Then $O$ is in particular a Dedekind domain Let for $$o\stackrel{i}{\to} O\to L$$ $f:=Spec(i):Spec(O)\to Spec(o)$ be the induced map between the [[nLab:spectrum|ring spectra]]. Let $p\in Spec(o)$ be a maximal prime ideal. Then the ideal $pO$ in $O$ has a unique decomposition $$pO=P_1^{e_1}\dots P_r^{e_r}$$ with different $P_i\in Spec(O)$ ## References * Jürgen Neukirch, algebraic number theory, I.§13.6{#NeuNum}