Let $S$ be a discrete valuation ring and $R\subset S$ be a proper subring (also a DVR). Assuming that $M$ and $N$ are the respective maximal ideals of $R$ and $S$ and that $N\cap R = M$, then the quotient field of $S$ is a proper subfield of the quotient field of $R$.
My Idea: Take $x\in S - R$ which must be non-zero. Then $(x)$ is an ideal of $S$. Since $(x)\neq 0$ and $S$ is a discrete valuation ring, then we must have that $(x)$ is an ideal of $S$.
I know this doesn't go very far, but there are a few hypothesis that are bizarre to me.
The requirement that $N\cap R = M$ sounds a bit like the requirements of the "going up" theorem [of which there seems to be quite a few versions :(] but since $R$ is integrally closed, $S$ cannot be integral over $R$. So this is paragraph I have all but completely dismissed.
To complete the story, I am stuck.