Let $R\subset S$ be two finitely generated integral domains over an algebraically closed field $k$. If $S$ is finite as $R$-module then $[L:K]<+\infty$, where $L$ and $K$ are the quotient fields of $S$ and $R$, respectively. Does the converse hold true? Any help is well accepted.
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1What about the integral domains $R = \mathbb{C}\left[t\right]$ and $S = \mathbb{C}\left[t,t^{-1}\right]$ over the field $k = \mathbb{C}$? Their quotient fields are the same. – darij grinberg Jan 12 '19 at 18:46
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Thanks. It was easy! – Vincenzo Zaccaro Jan 12 '19 at 18:48