p-adic order

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

$r=p_1^{{\nu }_{p_1}}{p}_2^{{\nu }_{p_2}}\cdots p_k^{{\nu }_{p_k}}=\prod _{i=1}^k{p}_i^{{\nu }_{p_i}},$

where $p_1<p_2<\cdots <p_k$ are primes and the ${\nu }_{p_i}$ are (unique) integers (considered to be 0 for all primes not occurring in r so that ${\nu }_{p_i}(r)={\nu }_{p_i}(a)-{\nu }_{p_i}(b)$).

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 $\mathbb{\mathbb{Q}}$_p (relative to a chosen prime number p), whereas the completion with the usual absolute value results in the field of real numbers $\mathbb{ℝ}$.

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 $\mathbb{\mathbb{Z}}$ is the function

${\nu }_p:\mathbb{\mathbb{Z}}\to \mathbb{\mathbb{N}}$

defined by

${\nu }_p(n)=\{\begin{array}{ll}\mathrm{m}\mathrm{a}\mathrm{x}\{k\in \mathbb{\mathbb{N}}:p^k\mid n\}& \text{if }n\ne 0\\ \mathrm{\infty }& \text{if }n=0,\end{array}$

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

For example, ${\nu }_3(-45)=2$ and ${\nu }_5(-45)=1$ since $|-45|=45=3^2\cdot 5^1$.

The notation $p^k\parallel n$ is sometimes used to mean $k={\nu }_p(n)$.

Rational numbers

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

${\nu }_p:\mathbb{\mathbb{Q}}\to \mathbb{\mathbb{Z}}$

defined by

${\nu }_p\left(\frac{a}{b}\right)={\nu }_p(a)-{\nu }_p(b).$

For example, ${\nu }_2\left(\frac{9}{8}\right)=-3$ and ${\nu }_3\left(\frac{9}{8}\right)=2$ since $\frac{9}{8}=\frac{3^2}{2^3}$.

Some properties are:

$\begin{array}{rl}{\nu }_p(m\cdot n)& ={\nu }_p(m)+{\nu }_p(n)\\ [5px]{\nu }_p(m+n)& \ge \min \{{\nu }_p(m),{\nu }_p(n)\}.\end{array}$

Moreover, if ${\nu }_p(m)\ne {\nu }_p(n)$, then

${\nu }_p(m+n)=\min \{{\nu }_p(m),{\nu }_p(n)\}$

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

p-adic absolute value

The p-adic absolute value on $\mathbb{\mathbb{Q}}$ is the function

$|\cdot {|}_p:\mathbb{\mathbb{Q}}\to {\mathbb{ℝ}}_{\ge 0}$

defined by

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

For example, $|-45{|}_3=\frac{1}{9}$ and $|\frac{9}{8}{|}_2=8.$

The p-adic absolute value satisfies the following properties.

Non-negativity	$|a{|}_p\ge 0$
Positive-definiteness	$|a{|}_p=0⟺a=0$
Multiplicativity	$|ab{|}_p=|a{|}_p|b{|}_p$
Non-Archimedean	$|a+b{|}_p\le \max (|a{|}_p,|b{|}_p)$

The symmetry $|-a{|}_p=|a{|}_p$ follows from multiplicativity $|ab{|}_p=|a{|}_p|b{|}_p$ and the subadditivity $|a+b{|}_p\le |a{|}_p+|b{|}_p$ from the non-Archimedean triangle inequality $|a+b{|}_p\le \max (|a{|}_p,|b{|}_p)$.

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

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

where the product is taken over all primes p and the usual absolute value, denoted $|x{|}_0$. This follows from simply taking the prime factorization: each prime power factor $p^k$ 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", 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 $\mathbb{\mathbb{Q}}$ with a (non-Archimedean, translation-invariant) metric

$d:\mathbb{\mathbb{Q}}\times \mathbb{\mathbb{Q}}\to {\mathbb{ℝ}}_{\ge 0}$

defined by

$d(x,y)=|x-y{|}_p.$

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

See also

• p-adic number
• Archimedean property
• Multiplicity (mathematics)
• Ostrowski's theorem