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Luroth's theorem states that if $k$ is a field and $L$ is a field extension of $k$ such that $k \subset L \subseteq k(X)$, then $L=k(f(X))$ for some $f(X) \in k(X) $ . My question is ; is there any analogous result if we replace field $k$ by a PID or a local PID i.e. DVR ?

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  • $\begingroup$ What is the analogue of $k(X)$? $\endgroup$ Mar 12, 2018 at 17:26
  • $\begingroup$ @LaurentMoret-Bailly: fraction field of $R[X]$, where $R$ is a DVR say, and $L$ is some ring extension of $R$ ... $\endgroup$
    – user111524
    Mar 12, 2018 at 17:49

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