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Let $k$ be an algebraically closed field and $(R, \mathfrak{m})$ a valuation ring in $k$ i.e. the field of fractions of $R$ is $k$. Then Mumford (Red Book, page 127) claimed that the residue field $R/\mathfrak{m}$ is also algebraically closed. Please help.

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Let $\bar{k}=R/\mathfrak{m}$ be the residue field of $R$. Fix a monic $\bar{f}(X)\in\bar{k}[X]$ of positive degree, and let $f\in R[X]$ be a monic lift of $\bar{f}$ (so also of positive degree). By assumption, $f$ has a root $\lambda$ in $k$. But valuation rings are integrally closed, so $\lambda$, being integral over $R$, is in $R$. Thus $\bar{\lambda}=\lambda+\mathfrak{m}$ is a root of $\bar{f}$ in $\bar{k}$, so $\bar{k}$ is algebraically closed.

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