Let $(R,\mathfrak m, \kappa) $ be a Discrete Valuation Ring (i.e., a local PID) containing a field $k\hookrightarrow \kappa$ such that $\kappa$ has transcendence degree $1$ over $k$. Let $K$ be the fraction field of $R$ and assume $K=k(a_1,...,a_n)$ for some $a_1,...,a_n\in K$. Also assume $\operatorname{tr.deg} K/k=1+\operatorname{tr.deg}\kappa/k$.
Then, how to show that $R$ is a finitely generated $k$-algebra ?
$R$ cannot be a finitely generated $k$-algebra because such an algebra posesses infinitely many prime ideals while $R$ has just two. However $R$ is a localization of a finitely generated $k$-algebra: let $\overline{x}$ be a transcendence basis of $\kappa/k$ and let $x\in R$ be a preimage of $\overline{x}$ under the residue map $R\rightarrow\kappa$. Then $k(x)\subset R$ and $K/k(x)$ is of transcendence degree equal to $1$. The elements $a_k$ can be chosen in $R$ - replace $a_k$ by $a_k^{-1}$ if necessary. Then $A:=k(x)[a_1,\ldots ,a_n]\subseteq R$ is a $1$-dimensional noetherian domain. Hence its integral closure $B$ in $K$ is a Dedekind domain contained in $R$ by the theorem of Krull-Akizuki. Therefore $R$ is a localization of $B$. Moreover $B$ is a finitely generated $k$-algebra.