My question is how to determine all valuation rings of the field $k(X)$ containg the field $k$.
I want to show that if $V$ is a valuation ring of the field $k(X)$ and $\neq k(X)$ then $V=k[X^{-1}]_{X^{-1}}$ or $V=k[X]_{f}$ for some irreducible polynomial $f$. I'm starting by: if $X\notin V$ then $X^{-1}\in V$, and I stuck to show that $V=k[X]_{X^{-1}}$. In the other case if $X\in V$, it is clear that $k[X]\subset V$, here how to construct $f$.
If $X^{-1}\in V$ and $X\notin V$, then $X^{-1}\in m$, where $m$ is the maximal ideal of $V$. We get $m\cap k[X^{-1}]=(X^{-1})$, and therefore $k[X^{-1}]_{(X^{-1})}\subseteq V$.
If $X\in V$, then $k[X]\subset V$, and $m\cap k[X]$ is a prime ideal of $k[X]$ generated by an irreducible polynomial $f$. Then we have $k[X]_{(f)}\subseteq V$.
Now note that in both cases $m$ lies over the maximal ideal of the subrings of $V$ which are also valuation rings, hence we must have equality.