Krull-Azikuzi theorem

Statement
Let:


 * $$R$$ be a Noetherian integral domain of Krull dimension 1
 * $$K$$ be the field of fractions of $$R$$
 * $$L$$ be a finite extension field of $$K$$
 * $$S$$ be a subring of $$L$$ that contains $$R$$

Then the following hold:


 * $$S$$ is Noetherian
 * The Krull dimension of $$S$$ is at most 1
 * Given any nonzero ideal of $$R$$, there are only finitely many ideals of $$S$$ containing that

In particular, the integral closure of $$R$$ in $$L$$ is Noetherian