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