# Sub-discrete valuation ring?

Let $A$ be a DVR in its field of fractions $F$, and $F'\subset F$ a subfield. Then is it true that $A\cap F'$ is a DVR in $F'$? I can see that it is a valuation ring of $F'$, but how do I I show, for example, that $A\cap F'$ is Noetherian?

Restrict the valuation corresponding to A to $F'$. Then the restriction is also a valuation with discreate value group( a subgroup of ring of integers) if it is non trivial. But if the value group is trivial, then it is field and therefore not a DVR.
This is not necessarily true. For instance, you could have $A = k[[t]]$ for a field $k$ and $F' \subset k((t))$ equal to $k$. Then $A \cap F' = F'$ and a field is often not considered a DVR.
But this is the only thing that can go wrong. In general, if a domain has a nontrivial valuation taking values in the nonnegative integers, such that the elements of valuation zero are units, then it is a DVR (and noetherianness follows --- for instance, one checks that the only ideals occur as ${x: x \ \mathrm{has \ valuation \ at \ least \ } n}.$)