Let $p$ be a positive prime number.  For every non-zero rational number $x$ there exists a unique integer $n$ such that

 $x=p^{n}\cdot\frac{u}{v}$

with some integers $u$ and $v$ indivisible by $p$.  We define

 $|x|_{p}:=\begin{cases}(\frac{1}{p})^{n}\quad\mathrm{when}\,\,x\neq 0,\\ 0\quad\mathrm{when}\,\,x=0,\end{cases}$

obtaining a non-trivial (http://planetmath.org/TrivialValuation) non-archimedean valuation, the so-called $p$-adic valuation

 $|\cdot|_{p}:\,\mathbb{Q}\to\mathbb{R}$

of the field $\mathbb{Q}$.

The value group of the $p$-adic valuation consists of all integer-powers of the prime number $p$.  The valuation ring of the valuation is called the ring of the p-integral rational numbers; their denominators, when reduced (http://planetmath.org/Fraction) to lowest terms, are not divisible by $p$.

The field of rationals has the 2-adic, 3-adic, 5-adic, 7-adic and so on valuations (which may be called, according to Greek, dyadic, triadic, pentadic, heptadic and so on).  They all are non-equivalent (http://planetmath.org/EquivalentValuations) with each other.

If one replaces the number $\frac{1}{p}$ by any positive $\varrho$ less than 1, one obtains an equivalent (http://planetmath.org/EquivalentValuations) $p$-adic valuation; among these the valuation with  $\varrho=\frac{1}{p}$  is sometimes called the normed $p$-adic valuation.  Analogously we can say that the absolute value is the normed archimedean valuation of $\mathbb{Q}$ which corresponds the infinite prime $\infty$ of $\mathbb{Z}$.

The product of all normed valuations of $\mathbb{Q}$ is the trivial valuation$|\cdot|_{\mathrm{tr}}$,  i.e.

 $\prod_{p\,\mathrm{prime}}|x|_{p}=|x|_{\mathrm{tr}}\quad\forall x\in\mathbb{Q}.$
Title p-adic valuation PadicValuation 2013-03-22 14:55:50 2013-03-22 14:55:50 pahio (2872) pahio (2872) 14 pahio (2872) Definition msc 13A18 $p$-adic valuation IndependenceOfPAdicValuations IntegralElement OrderValuation StrictDivisibility $p$-integral rational number normed $p$-adic valuation normed archimedean valuation dyadic valuation triadic valuation pentadic valuation heptadic valuation