It's from a statement in Algebraic Number Theory by Neukirch, page 92. Example 5 "One can show..."
Let ${o}$ be a one-dimensional noetherian integral domain and $\tilde{o}$ be its normalization, which is its integral closure, a dedekind domain. Hence we can decompose any prime $\frak{p}$ into primes in $\tilde{o}$. He states that $\frak{p}$ is regular point, which means the localization $o_{\frak{p}}$ is a discrete valuation ring if and only if we have $r=1, e=1$, $f=(\tilde{o}/\beta:o/\frak{p})$ in the decomposition of $\frak{p}$.
$r$ is the number of primes above $\frak{p}$, call it $\beta$ since $r=1$; $e$ is the ramification and $f$ is the degree of the field extension.
Now I have solved the necessity part, but I don't know where to start to use $r=e=f=1$ to deduce that $o_{\frak{p}}$ is a D.V.R and I don't know if there were any easy ways to approach it. The only thing we need is to show that $o_{\frak{p}}$ is integrally closed (by Prop 9.2 in Atiyah).
Thanks a lot if you can help me!
We have an injective ring map $o \to \tilde{o}$ corresponding to the inclusion of $o$ into its integral closure. Now, by hypothesis, $\mathfrak{p}\tilde{o} = \mathfrak{q}$ is a prime ideal of $\tilde{o}$ so that $\mathfrak{q} \cap \tilde{o} = \mathfrak{p}$.
Now consider the inclusion $o_{\mathfrak{p}} \to \tilde{o}_{\mathfrak{q}}$. We will show this is bijective and hence an isomorphism.
This is an injective map of $o_{\mathfrak{p}}$-modules, so to conclude, it suffices to show that $1 $ generates $\tilde{o}_{\mathfrak{q}}$ as an $o_{\mathfrak{p}}$-module. This then follows from Nakayama's lemma, since the image of $1$ generates $\tilde{o}_{\mathfrak{q}}/{\mathfrak{p}\tilde{o}_{\mathfrak{q}}} = \tilde{o}_{\mathfrak{q}}/\mathfrak{q}\tilde{o}_{\mathfrak{q}}$ as an $o_{\mathfrak{p}}/\mathfrak{p}o_{\mathfrak{p}}$-module, due to the fact that $f = 1$.