I don't understand ii)$\Longleftrightarrow$iii), but I know (9.2),(4.8),and(3.11). Could someone give me the complete proof?
DEDEKIND DOMAINS
Theorem 9.3. Let A be a Noetherian domain of dimension one. Then the
following are equivalent:
i) A is integrally closed;
ii) Every primary ideal in A is a prime power;
iii) Every local ring $A_\mathfrak p $($\mathfrak p\neq 0$)is a discrete valuation ring.
Proof. ii)$\Longleftrightarrow$iii). Use (9.2) and the fact that primary ideals and powers of ideals behave well under localization:(4.8), (3.11).
I'll give some hints, and hope you can complete the details by yourself.
Let's see ii)$\Rightarrow$iii): if $A$ is a Noetherian domain of dimension one, the same happens for $A_{\mathfrak{p}}$ with $\mathfrak{p}\neq 0$ a prime ideal. We will see that $A_{\mathfrak{p}}$ is a DVR using (9.2, v). Take any non-zero ideal $\mathfrak{a}$ of $A_{\mathfrak{p}}$, and let $\mathfrak{b}$ be its contraction to $A$. Let $\mathfrak{a}=\Pi \mathfrak{q}_i$ be a decomposition into primary ideals of $A_{\mathfrak{p}}$. Do you see what happens if we take the contraction to $A$ of this decomposition, and use that every primary ideal of $A$ is a power of a prime?
Now iii)$\Rightarrow$ ii): take $\mathfrak{q}$ a $\mathfrak{p}$-primary ideal of $A$ (we can assume $\mathfrak{p}\neq 0$). Now, $S^{-1}\mathfrak{q}$ is $S^{-1}\mathfrak{p}$-primary; but $A_{\mathfrak{p}}$ is a DVR by hypothesis, so using again (9.2, v), $S^{-1}\mathfrak{q}=(S^{-1}\mathfrak{p})^{n}$ for some $n>0$. Now take the contraction to $A$.