Consider a quadratic field $\mathbb{Q}\sqrt{-15}$ and $R$ as its algebraic integers, let $\mathfrak{p}$ = $(2,\frac{1+\sqrt{-15}}{2})$, then it is rather not difficult to show $\mathfrak{p}$ is a prime ideal which is not principal. One wishes to prove that $R_\mathfrak{p}$ is a local PID.
I am aware that we have two powerful theorems to deal with such situation.
There is a one-to-one order-preserving correspondence between the prime ideals of the localization $R_T$ and the prime ideals of $R$ which are disjoint from $T$. This actually guarantees $R_\mathfrak{p}$ is the unique maximal ideal as $\mathfrak{p}$ is trivially the unique maximal ideal in $R-\mathfrak{p}$.
If $A$ is a PID, and $\mathfrak{p}$ a prime ideal in A, then the localization $A_\mathfrak{p}$ is also a PID.
However in this situation 1 applies and 2 doesn't apply. I am stuck showing $R_\mathfrak{p}$ is a PID.
Here are several ideas that cross my mind:
What is the structure of prime ideals in R? If one could figure it out, then by theorem 1 we know the primes in $R_\mathfrak{p}$ are, and we only need to show each of them is generated by one element. But classifying prime ideals in $R$ seems to be no easy task. And I don't have a lot of tools (except for norm maybe...?)
modify the proof of 2 somehow...?
*I am a noob in abstract algebra and commutative algebra so any answer/remarks would be of extreme help. I am currently going through Dummit & Foote's Abstract Algebra.