P-adic valuation

In number theory, for a given prime number p, the p-adic order or p-adic valuation of a non-zero integer n is the highest exponent $\nu$ such that $p^{\nu }$ divides n. The p-adic valuation of 0 is defined to be infinity. The p-adic valuation is commonly denoted $\nu _{p}(n)$. If ⁠n/d⁠ is a rational number in lowest terms, so that n and d are coprime, then $\nu _{p}({\tfrac {n}{d}})$ is equal to $\nu _{p}(n)$ if p divides n, or $-\nu _{p}(d)$ if p divides d, or to 0 if it divides neither. The most important application of the p-adic order is in constructing the field of p-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.

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

Integers

Let p be a prime number. The p-adic order or p-adic valuation for ℤ is the function $\nu _{p}:\mathbb {Z} \to \mathbb {N}$ defined by

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

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

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

Rational numbers

The p-adic order can be extended into the rational numbers as the function $\nu _{p}:\mathbb {Q} \to \mathbb {Z}$ defined by

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

For example, $\nu _{5}({\tfrac {9}{10}})=-1$.

Some properties are:

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

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

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

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

p-adic absolute value

The p-adic absolute value on ℚ is defined as |·|_p : ℚ → ℝ

$|x|_{p}={\begin{cases}p^{-\nu _{p}(x)}&{\text{if }}x\neq 0\\0&{\text{if }}x=0.\end{cases}}$

For example, $|45|_{3}={\tfrac {1}{9}}$ and $|{\tfrac {9}{10}}|_{5}=5$.

The p-adic absolute value satisfies the following properties.

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

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

A metric space can be formed on the set ℚ with a (non-Archimedean, translation-invariant) metric defined by d : ℚ × ℚ → ℝ

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

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.

The choice of base p in the formula makes no difference for most of the properties, but results in the product formula:

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

where the product is taken over all primes p and the usual absolute value (Archimedean norm), 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.

See also

• Archimedean property
• Multiplicity (mathematics)
• Ostrowski's theorem