Dedekind rings $R \subset R'$ where $R'_{S}$ is not finitely generated over $R_S$ but their quotient fields form a finite extension?

Question

Do there exist Dedekind rings $R \subset R'$, where $R'$ integral over $R$, and corresponding quotient fields $K \subset L$ and some prime ideal $\mathfrak{p} \subset R$ where if $S := R\setminus \mathfrak{p}$ such that:

(1) $R'_S$ is not finitely generated over $R_S$

and

(2) $[L : K] < \infty$?

(For motivation, I am reading Gerald Janusz's Algebraic Number Fields in which he states a theorem that in the above setup if $R'_S$ is finitely generated over $R_S$, then we obtain the equation $[L : K] = \sum e_if_i$ corresponding to the ramification index and relative degree associated to the factors of $\mathfrak{p}$.)

Answer 1

$R_S$ is a principal ideal domain ( local ring too). So if $R'_S$ is finitely generated ( as a module), then it's free. Moreover, one checks easily that $R'_S$ is always the integral closure of $R_S$ in $L$. So $R'_S$ is free of rank $[L\colon K]$. Now, if you want to search for $R'$ that are not finitely generated over $R$, you need to look at non-separable extensions, since the integral closure in a finite separable extension is a finitely generated extension.

Necessarily for this is $R'$ is not a finite $R$ module.

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