Another result I would really appreciate some help with:
Suppose $R$ is a DVR and let $K$ be its field of fractions. Let $L$ be a finite extension of $L$. Prove that any valuation domain inside of $L$ containing $R$ must also be a DVR.
This is exercise 11.2 from Matsumura's Commutative Ring Theory. I suppose it uses the Krul-Akizuki theorem but I don't see it. Thank you!