Discrete valuation arising from localizations of $R[[t]]$ Let $R$ be a DVR and consider the ring of power series $R[[t]]$. Now let $\mathfrak p\subset R[[t]]$ be a prime ideal of height $1$.

Why the localization $R[[t]]_{\mathfrak p}$ is again a DVR?


The ring $R[[t]]$ is local of dimension $2$. Furthermore, its maximal ideal is generated by two elements, a uniformizing parameter of $R$ and $t$. Thus, $R[[t]]$ is regular. A localization of a regular ring is regular, and a regular local ring of dimension 1 is a DVR.
