Fulton, Algebraic curves 4.8 Let $V=\mathbb{P}^1$, with corresponding coordinate ring $\Gamma_h(V)=k[X,Y]$. Let $t=X/Y\in k(V)$. The question is following :

*

*Show $k(V)=k(t)$.

*There is a 1-1 correspondence between points of $\mathbb{P}^1$ and the DVR's with quotient field $k(V)$ that contain $k$.

I could solve 1, but I have no idea for 2. I guess desired DVR is $O_p(V)$ (local ring of V at p), but I can't show why the maximal ideal $\mathfrak{m}_p(V):=\{f/g : g(p)\neq0 , f(p)=0\}$ is principal ideal, and conversely why such local rings of V at p are all discrete valuation ring with quotient field $k(V)$ containing $k$. Thank you for any helps.
 A: Fulton fixes $k$ algebraically closed after section 3, so let $k$ be algebraically closed.
Cover $\Bbb P^1_k$ with the standard affine charts $D_+(x)\cong \text{Spec}(k[y/x])$ and $D_+(y)\cong\text{Spec}(k[x/y])$. Then the stalk at any closed point of $\Bbb P^1_k$ is isomorphic to $k[s]_{\mathfrak{p}}$ where $\mathfrak{p}$ is a maximal ideal (hence has height $1$, in which case $\text{dim}(k[s]_\mathfrak{p})=1$).
A discrete valuation ring is another name for a regular local ring of dimension $1$, so $k[s]_\mathfrak{p}$ is a DVR. Fix $t=y/x$. If $A$ is a DVR such that $k\subset A\subset k(t)$, then either $t\in A$ or $t^{-1}\in A$ (and hence $k[t]\subset A$ or $k[t^{-1}]\subset A$). In the case that $k[t]\subset A$, contracting the maximal ideal $\mathfrak{m}\subset A$, we see that $\mathfrak{m}\cap k[t]$ is maximal, i.e. generated by some $(t-a)$, and since all non-zero elements of $A\backslash\mathfrak{m}$ are invertible, $k[t]_{(t-a)}\subseteq A$, and these have the same fraction field, so they are equal.
Returning to $\Bbb P^1_k$, let $a\ne 0$ and consider $(bx-ay)\in\Bbb P^1_k$, which belongs to $D_+(x)$, and corresponds to $(a(y/x)-b)\in \text{Spec}(k[y/x])$. The stalk at this point is isomorphic to $k[t]_{(at-b)}$ with maximal ideal $(at-b)k[t]_{(at-b)}$. This example gives you all the DVRs with fraction field $k(t)$ which contain $k$ except for the one corresponding to $a=0$ (which corresponds to $(x)\in D_+(y)$ and hence to the discrete valuation ring $k[x/y]_{(x/y)}\cong k[t^{-1}]_{(t^{-1})}\subset k(t)$).
