p-adic valuationPAdicValuation2005-01-04 14:31:432008-05-23 11:37:09DefinitionOstrowski's valuation theorem$p$-integral rational numbernormed $p$-adic valuationnormed archimedean valuationdyadic valuationtriadic valuationpentadic valuationheptadic valuation% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $p$ 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 \PMlinkname{non-trivial}{TrivialValuation} non-archimedean valuation, the so-called $p$-{\em 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 {\em p-integral rational numbers}; their denominators, when \PMlinkname{reduced}{Fraction} to lowest terms, are not divisible by $p$.
The field of rationals has the {\em 2-adic, 3-adic, 5-adic, 7-adic} and so on valuations (which may be called, according to Greek, {\em dyadic, triadic, pentadic, heptadic} and so on).\, They all are \PMlinkname{non-equivalent}{EquivalentValuations} with each other.
If one replaces the \PMlinkescapetext{base} number $\frac{1}{p}$ by any positive \PMlinkescapetext{constant} $\varrho$ less than 1, one obtains an \PMlinkname{equivalent}{EquivalentValuations} $p$-adic valuation; among these the valuation with\, $\varrho = \frac{1}{p}$\, is sometimes called the {\em 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.
$$\prod_{p\,\mathrm{prime}}|x|_p = |x|_\mathrm{tr} \quad
\forall x\in\mathbb{Q}.$$