Let be given a domain $R$ (that can be supposed to be integrally closed if this can help), and $\varphi$ an homomorphism of $R$ into a field $F$. $\varphi$ extends uniquely to a homomorphism $\varphi'$ of $R_I$ to $F$, where $I=\ker(\varphi)$, and $\varphi'$ extends itself to a place $\tilde \varphi$ of $K={\rm quot}(R)$ to $\tilde F \cup \infty$, where $\tilde F$ is an algebraic extension of $F$. We can assume without loss of generality that $F={\rm Im}(\varphi') = {\rm quot}({\rm Im}(\varphi))$ and that $\tilde F$ is the finite image of $\tilde\varphi$. What is known about the extension $\tilde F/F$ (in particular is it trivial ?), and what is known about the number of ways to extend $\varphi'$ to a place of $K$ ?

Equivalently, if $O$ is the localization of a domain at a prime ideal, what is known about the number of ways to extend it to a valuation ring $\tilde O$ of ${\rm quot}(O)$ such that its maximal ideal is contained in the maximal ideal of $\tilde O$ ?

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