Let $V = \mathbb A^1(k)$ ($k$ is an algebraically closed field), $\Gamma(V) = k[X]$ and let $K = k(V) = k(X)$. Prove that for each $a \in k = V$, $\mathcal{O}_a(V) := \{f\in K(V): f$ is defined at $a\}$ is a DVR with uniformizing parameter $t = X - a$.
Discrete Valuation Rings
3
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algebraic-geometry
ring-theory
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0I did my best for the LaTeX format. Please define Oa(V). – 2012-11-06
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0Oa(V):={f∈K(V); f is defined at a} – 2012-11-06
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1Welcome to Math.Se! Is it homework? What did you try to do? Where did you get stuck? – 2012-11-06
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0Use the fact that $F(x)\in\Gamma(V)$ has a root at $a$ iff $x-a~\vert~F(x).$ – 2012-11-06