Let p be a prime number.

Integers edit

The p-adic valuation of an integer ${\displaystyle n}$  is defined to be

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

where ${\displaystyle \mathbb {N} _{0}}$  denotes the set of natural numbers (including zero) and ${\displaystyle m\mid n}$  denotes divisibility of ${\displaystyle n}$  by ${\displaystyle m}$ . In particular, ${\displaystyle \nu _{p}}$  is a function ${\displaystyle \nu _{p}\colon \mathbb {Z} \to \mathbb {N} _{0}\cup \{\infty \}}$ .^[2]

For example, ${\displaystyle \nu _{2}(-12)=2}$ , ${\displaystyle \nu _{3}(-12)=1}$ , and ${\displaystyle \nu _{5}(-12)=0}$  since ${\displaystyle |{-12}|=12=2^{2}\cdot 3^{1}\cdot 5^{0}}$ .

The notation ${\displaystyle p^{k}\parallel n}$  is sometimes used to mean ${\displaystyle k=\nu _{p}(n)}$ .^[3]

If ${\displaystyle n}$  is a positive integer, then

${\displaystyle \nu _{p}(n)\leq \log _{p}n}$ ;

this follows directly from ${\displaystyle n\geq p^{\nu _{p}(n)}}$ .

Rational numbers edit

The p-adic valuation can be extended to the rational numbers as the function

${\displaystyle \nu _{p}:\mathbb {Q} \to \mathbb {Z} \cup \{\infty \}}$ ^[4][5]

defined by

${\displaystyle \nu _{p}\left({\frac {r}{s}}\right)=\nu _{p}(r)-\nu _{p}(s).}$ 

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

Some properties are:

${\displaystyle \nu _{p}(r\cdot s)=\nu _{p}(r)+\nu _{p}(s)}$ 

${\displaystyle \nu _{p}(r+s)\geq \min {\bigl \{}\nu _{p}(r),\nu _{p}(s){\bigr \}}}$ 

Moreover, if ${\displaystyle \nu _{p}(r)\neq \nu _{p}(s)}$ , then

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

where ${\displaystyle \min }$  is the minimum (i.e. the smaller of the two).

The p-adic absolute value (or p-adic norm,^{[citation needed]} though not a norm in the sense of analysis) on ${\displaystyle \mathbb {Q} }$  is the function

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

defined by

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

Thereby, ${\displaystyle |0|_{p}=p^{-\infty }=0}$  for all ${\displaystyle p}$  and for example, ${\displaystyle |{-12}|_{2}=2^{-2}={\tfrac {1}{4}}}$  and ${\displaystyle {\bigl |}{\tfrac {9}{8}}{\bigr |}_{2}=2^{-(-3)}=8.}$ 

The p-adic absolute value satisfies the following properties.

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

From the multiplicativity ${\displaystyle |rs|_{p}=|r|_{p}|s|_{p}}$  it follows that ${\displaystyle |1|_{p}=1=|-1|_{p}}$  for the roots of unity ${\displaystyle 1}$  and ${\displaystyle -1}$  and consequently also ${\displaystyle |{-r}|_{p}=|r|_{p}.}$  The subadditivity ${\displaystyle |r+s|_{p}\leq |r|_{p}+|s|_{p}}$  follows from the non-Archimedean triangle inequality ${\displaystyle |r+s|_{p}\leq \max \left(|r|_{p},|s|_{p}\right)}$ .

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:

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

where the product is taken over all primes p and the usual absolute value, denoted ${\displaystyle |r|_{0}}$ . 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.

A metric space can be formed on the set ${\displaystyle \mathbb {Q} }$  with a (non-Archimedean, translation-invariant) metric

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

defined by

${\displaystyle d(r,s)=|r-s|_{p}.}$ 

The completion of ${\displaystyle \mathbb {Q} }$  with respect to this metric leads to the set ${\displaystyle \mathbb {Q} _{p}}$  of p-adic numbers.

• p-adic number
• Valuation (algebra)
• Archimedean property
• Multiplicity (mathematics)
• Ostrowski's theorem
• Legendre's formula