p-adic valuation

Fix a prime $p$. For a nonzero integer $m$, the $p$-adic valuation $v_p(m)$ is the unique nonnegative integer $r$ such that $p^r\mid m$ but $p^{r+1}\nmid m$. Set $v_p(0)=+\infty$, and extend to nonzero rational numbers by

It satisfies $v_p(xy)=v_p(x)+v_p(y)$ and $v_p(x+y)\ge\min\{v_p(x),v_p(y)\}$. The associated absolute value is $|0|_p=0$ and $|x|_p=p^{-v_p(x)}$ for $x\ne0$; it defines the $p$-adic topology on $\mathbb Q$.