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[parent] p-adic valuation (Definition)

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

$\displaystyle x = p^n\cdot\frac{u}{v}$
with some integers $ u$ and $ v$ indivisible by $ p$. 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 $ p$-adic valuation
$\displaystyle \vert\cdot\vert _p:\,\mathbb{Q} \to \mathbb{R}$
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 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 with each other.

If one replaces the base number $ \frac{1}{p}$ by any positive constant $ \varrho$ less than 1, one obtains an equivalent $ 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 $ \vert\cdot\vert _\mathrm{tr}$, i.e.

$\displaystyle \prod_{p\,\mathrm{prime}}\vert x\vert _p = \vert x\vert _\mathrm{tr} \quad \forall x\in\mathbb{Q}.$



"p-adic valuation" is owned by pahio.
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See Also: independence of $p$-adic valuations, integral element, order valuation, strict divisibility

Other names:  $p$-adic valuation
Also defines:  $p$-integral rational number, normed $p$-adic valuation, normed archimedean valuation, dyadic valuation, triadic valuation, pentadic valuation, heptadic valuation

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Cross-references: trivial valuation, product, infinite prime, absolute value, dyadic, rationals, divisible, lowest terms, denominators, rational numbers, integral, ring, valuation ring, value group, field, valuation, non-archimedean, integer, rational number, prime number, positive
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This is version 10 of p-adic valuation, born on 2005-01-04, modified 2007-03-27.
Object id is 6619, canonical name is PAdicValuation.
Accessed 6769 times total.

Classification:
AMS MSC13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)

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