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Let $k$ be field and $K=k(x)$ the field of rational function functions over $k$. Suppose that $p(x)\in k[x]$ is an irreducible polynomial. Define a map $v:K\to\mathbb Z$ by $v(p^r\frac{m}{n})=r$ where $m(x)$, $n(x)$ are polynomials relatively prime to each other and to $p(x)$. I can show $v$ is discrete valuation, but I can not find the valuation ring $O_v$ and units in $O_v$.

I know definition $O_v=\{x\in K: v (x)\geq 0\}$.

user26857
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corcia candy
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  • It is simply the ring of rational functions with denominator not divisible by $p$ (when in irreducible form). Units are rational functions with neither numerator nor denominator divisible by $p$. – Bernard Jan 07 '16 at 10:15
  • @Bernard I can see it from defn but I couldnt find that which ring is isomorphic $O_k $ – corcia candy Jan 07 '16 at 11:07
  • What else do you want to say than ‘the localisation of $k[x]$ at the prime ideal $pk[x]$’? – Bernard Jan 07 '16 at 12:34

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