Q:

Finite Extension of Integral Domains.

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.

A:

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?).

