PlanetMath: p-adic valuation

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p-adic valuation

(Definition)

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

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

with some integers

and

indivisible by

. We define

$\begin{displaymath}\vert x\vert _p := \begin{cases} (\frac{1}{p})^n \quad \mathr... ... \,\, x \neq 0, \ 0 \quad \mathrm{when} \,\, x=0, \end{cases}\end{displaymath}$

obtaining a non-trivial non-archimedean valuation, the so-called

-adic valuation

$\displaystyle \vert\cdot\vert _p:\,\mathbb{Q} \to \mathbb{R}$

of the field $\mathbb{Q}$ .

The value group of the -adic valuation consists of all integer-powers of the prime number . The valuation ring of the valuation is called the ring of the -integral rational numbers; their denominators, when reduced to lowest terms, are not divisible by .

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 with each other.

If one replaces the base number $\frac{1}{p}$ by any positive constant $\varrho$ less than 1, one obtains an equivalent -adic valuation; among these the valuation with $\varrho = \frac{1}{p}$ is sometimes called the normed -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 $\mathbb{Z}$ .