Is the localization of a polynomial ring at a prime ideal a valuation ring if and only if the prime ideal is principal?
It seems to me that it is the case, and here's my work on the question in the form of an argument:
Suppose $R$ is an integral domain, let $\mathfrak p$ be a principal prime ideal of $R[x_1,\dots,x_n]$ generated by $p$. We note $\text{Frac}(R[x_1,\dots,x_n]_{\mathfrak p}) = R(x_1,\dots,x_n)$. Suppose $f \in R(x_1, \dots, x_n)$. If both the numerator and the denominator are divided by $p$, then $f$ is equivalent to some other $f'$ s.t. $p$ divides either only the numerator, or only the denominator. Clearly then, $f'$ or $f'^{-1}$ is in $R[x_1,\dots, x_n]_{\mathfrak p}$
Conversely, consider the localization of a polynomial ring at a prime ideal $\mathfrak p$ that is not principal. Then $\mathfrak p$ has at least two generators, $a,b$. But $a/b \in R(x_1,\dots, x_n)$, yet neither $a/b$ nor $b/a$ are in $R[x_1,\dots,x_n]_{\mathfrak p}$, establishing the desired proposition.