View source for P-adic valuation

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Fix a prime number $p$.  The $p$-adic [[valuation]] (or order) $\nu_p({\cdot})$ on the field of [[rational number]]s is defined by $\nu(a/b) = r$ where $a,b$ are integers and $a/b = p^r.a'/b'$ with $a',b'$ coprime to $p$; set $\nu_p(0) = \infty$.  The $p$-adic [[Norm on a field|norm]] $\Vert{\cdot}\Vert_p$ is defined by $\Vert x \Vert_p = p^{-\nu_p(x)}$, with $\Vert 0 \Vert_p = 0$.  

The $p$-adic order satisfies
* $\nu_p(x y) = \nu_p(x) + \nu_p(y)$;
* $\nu_p(x+y) \ge \min\{\nu_p(x),\nu_p(y)\}$, with equality if the two terms are unequal.

The $p$-adic norm correspondingly satisfies
* $\Vert xy \Vert_p = \Vert x \Vert_p \Vert y \Vert_p$;
* $\Vert x+y \Vert_p \le \max\{\Vert x \Vert_p, \Vert y \Vert_p\}$.

The $p$-adic norm is thus an [[ultrametric]] norm.  The [[p-adic number|$p$-adic numbers]] are the completion of the rationals with respect to the $p$-adic norm, and the $p$-adic valuation extends to the field of $p$-adic numbers.

Ostrowki's theorem states that the only norms on the field of rational numbers are the $p$-adic norms and the usual absolute value.

====References====
* Cassels, J.W.S. "Local fields" London Mathematical Society Student Texts '''3''' Cambridge University Press (1986)  ISBN 0-521-31525-5 {{ZBL|0595.12006}}