p-adic valuation

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

$${|x|}_p:=\{\begin{array}{cc}{(\frac{1}{p})}^n\hspace{1em}\mathrm{when}x\ne 0,\hfill & \\ 0\hspace{1em}\mathrm{when}x=0,\hfill & \end{array}$$

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

$$|\cdot {|}_p:\mathbb{Q}\to ℝ$$

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 $\mathrm{\infty }$ of $\mathbb{Z}$.

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

$$\prod _{p\mathrm{prime}}{|x|}_p={|x|}_{\mathrm{tr}}\mathit{\hspace{1em}}\forall x\in \mathbb{Q}.$$