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In basic number theory, for a given prime number p, the p-adic order of a positive integer n is the highest exponent ${\displaystyle \nu _{p}}$ such that ${\displaystyle p^{\nu _{p}}}$ divides n. This function is easily extented to positive rational numbers r = a/b by

${\displaystyle r=p_{1}^{\nu _{p_{1}}}p_{2}^{\nu _{p_{2}}}\cdots p_{k}^{\nu _{p_{k}}}=\prod _{i=1}^{k}p_{i}^{\nu _{p_{i}}},}$

where ${\displaystyle p_{1} are primes and the ${\displaystyle \nu _{p_{i}}}$ are (unique) integers (considered to be 0 for all primes not occurring in r so that ${\displaystyle \nu _{p_{i}}(r)=\nu _{p_{i}}(a)-\nu _{p_{i}}(b)}$).

This p-adic order constitutes an (additively written) valuation, the so-called p-adic valuation, which when written multiplicatively is an analogue to the well-known usual absolute value. Both types of valuations can be used for completing the field of rational numbers, where the completion with a p-adic valuation results in a field of p-adic numbers p (relative to a chosen prime number p), whereas the completion with the usual absolute value results in the field of real numbers .[1]

Distribution of natural numbers by their 2-adic order, labeled with corresponding powers of two in decimal. Zero always has an infinite order

## Definition and properties

Let p be a prime number.

### Integers

The p-adic order or p-adic valuation for is the function

${\displaystyle \nu _{p}:\mathbb {Z} \to \mathbb {N} }$ [2]

defined by

${\displaystyle \nu _{p}(n)={\begin{cases}\mathrm {max} \{v\in \mathbb {N} :p^{v}\mid n\}&{\text{if }}n\neq 0\\\infty &{\text{if }}n=0,\end{cases}}}$

where ${\displaystyle \mathbb {N} }$  denotes the natural numbers.

For example, ${\displaystyle \nu _{3}(-45)=2}$  and ${\displaystyle \nu _{5}(-45)=1}$  since ${\displaystyle |{-45}|=45=3^{2}\cdot 5^{1}}$ .

### Rational numbers

The p-adic order can be extended into the rational numbers as the function

${\displaystyle \nu _{p}:\mathbb {Q} \to \mathbb {Z} }$ [3]

defined by

${\displaystyle \nu _{p}\left({\frac {a}{b}}\right)=\nu _{p}(a)-\nu _{p}(b).}$ [4]

For example, ${\displaystyle \nu _{2}{\bigl (}{\tfrac {9}{8}}{\bigr )}=-3}$  and ${\displaystyle \nu _{3}{\bigl (}{\tfrac {9}{8}}{\bigr )}=2}$  since ${\displaystyle {\tfrac {9}{8}}={\tfrac {3^{2}}{2^{3}}}}$ .

Some properties are:

{\displaystyle {\begin{aligned}\nu _{p}(m\cdot n)&=\nu _{p}(m)+\nu _{p}(n)\\[5px]\nu _{p}(m+n)&\geq \min {\bigl \{}\nu _{p}(m),\nu _{p}(n){\bigr \}}.\end{aligned}}}

Moreover, if ${\displaystyle \nu _{p}(m)\neq \nu _{p}(n)}$ , then

${\displaystyle \nu _{p}(m+n)=\min {\bigl \{}\nu _{p}(m),\nu _{p}(n){\bigr \}}}$

where min is the minimum (i.e. the smaller of the two).

## p-adic absolute value

The p-adic absolute value on is the function

${\displaystyle |\cdot |_{p}\colon \mathbb {Q} \to \mathbb {R} _{\geq 0}}$

defined by

${\displaystyle |r|_{p}=p^{-\nu _{p}(r)}.}$ [4]

For example, ${\displaystyle |{-45}|_{3}={\tfrac {1}{9}}}$  and ${\displaystyle {\bigl |}{\tfrac {9}{8}}{\bigr |}_{2}=8.}$

The p-adic absolute value satisfies the following properties.

 Non-negativity ${\displaystyle |a|_{p}\geq 0}$ Positive-definiteness ${\displaystyle |a|_{p}=0\iff a=0}$ Multiplicativity ${\displaystyle |ab|_{p}=|a|_{p}|b|_{p}}$ Non-Archimedean ${\displaystyle |a+b|_{p}\leq \max \left(|a|_{p},|b|_{p}\right)}$

The symmetry ${\displaystyle |{-a}|_{p}=|a|_{p}}$  follows from multiplicativity ${\displaystyle |ab|_{p}=|a|_{p}|b|_{p}}$  and the subadditivity ${\displaystyle |a+b|_{p}\leq |a|_{p}+|b|_{p}}$  from the non-Archimedean triangle inequality ${\displaystyle |a+b|_{p}\leq \max \left(|a|_{p},|b|_{p}\right)}$ .

The choice of base p in the exponentiation ${\displaystyle p^{-\nu _{p}(r)}}$  makes no difference for most of the properties, but supports the product formula:

${\displaystyle \prod _{0,p}|x|_{p}=1}$

where the product is taken over all primes p and the usual absolute value, denoted ${\displaystyle |x|_{0}}$ . This follows from simply taking the prime factorization: each prime power factor ${\displaystyle p^{k}}$  contributes its reciprocal to its p-adic absolute value, and then the usual Archimedean absolute value cancels all of them.

The p-adic absolute value is sometimes referred to as the "p-adic norm",[citation needed] although it is not actually a norm because it does not satisfy the requirement of homogeneity.

A metric space can be formed on the set with a (non-Archimedean, translation-invariant) metric

${\displaystyle d\colon \mathbb {Q} \times \mathbb {Q} \to \mathbb {R} _{\geq 0}}$

defined by

${\displaystyle d(x,y)=|x-y|_{p}.}$

The completion of with respect to this metric leads to the field p of p-adic numbers.