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I want to show that only discrete valuation rings with quotient field as $k(x)$ containing $k$ are: $\mathcal{O_{a} (\mathbb{A^{1}})}$ for each $a \in k$ and $\mathcal{O_{\infty}}$; the former is the set of rational functions on $\mathbb{A^{1}}$ (affine 1-space, that is field $k$ here) that are defined at $a \in k$, it is a discrete valuation ring with uniformizing parameter $x-a$ and the latter is the ring $$ \left\{\frac{F}{G} \in k(x) \mid \deg(G) \geq \deg(F) \right\} $$ with $\frac{1}{x}$ as its uniformizing parameter.

My idea was to first observe that if $S$ is any DVR, then it cannot be clearly field of quotients $k(x)$, since in the book (Fulton, Algebraic Curves) we have not defined them as fields. So, $S\subset k(x)$.

It will contain the ring $k[x]$. Now I will use a previous exercise that says that "If $R$ is a DVR with quotient field $K$ and $m$ as its maximal ideal then for $z\in K, z \notin R$, we must have $z^{-1} \in m$." and another that says that

"Further if $R\subset S\subset K$ and $S $ is also a DVR, and the maximal ideal of $S$ contains $m$ then $S =R$."

But I don't know how I can start.

Any hint would be appreciated, thanks!

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  • $\begingroup$ Is $k$ algebraically closed? Otherwise, I don't think the statement is true. (You'd have to restrict to discrete valuations that are trivial on $k$.) $\endgroup$ – André 3000 Feb 11 at 7:55
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    $\begingroup$ It is algebraically closed. Also, thanks for editing I was not able to correct that editing error so left it as it is. $\endgroup$ – Mojojojo Feb 11 at 8:03
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Let me start by pointing out the a DVR $R\subset k(x)$ may not contain $k[x]$, and in fact you $\mathscr{O}_\infty$ doesn't. Here's an outline of how to prove the result: (1) using the second exercise you mention (about maximality of DVRs), prove that a DVR R that does contain $k[x]$ must be the localization of $k[x]$ at some maximal ideal, and so isomorphic to some $\mathscr{O}_a (\mathbb{A}^1)$ (hint: if $\mathfrak{m}$ is the maximal ideal of $R$, what is $\mathfrak{m}\cap k[x]$?) (2) by the first exercise you cite, if $R$ does not contain $k[x]$, it must contain $x^{-1}$; use maximality again to prove that such a DVR must be $\mathscr{O}_\infty$.

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