Let $D\subset E$ (integral domains), with fraction fields $k\subset K $.
Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated.
My question is:
$[K:k]$ is finite?
Thank you all.
2026-03-25 18:55:00.1774464900
Finite Extension of Integral Domains.
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Suppose that $E$ is a finitely generated $D$-module. Say $E=De_1+\cdots+De_n$. Let's show that the field extension $k\subset K$ is finite. It is clearly algebraic (why?) and of finite type: $K=k(e_1,\dots,e_n)$ (why?).