p-adic valuation – Wikipedia

In number theory, the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n.
It is denoted

.
Equivalently,

is the exponent to which

appears in the prime factorization of

.

The p-adic valuation is a valuation and gives rise to an analogue of the usual absolute value.
Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers

, the completion of the rational numbers with respect to the

-adic absolute value results in the p-adic numbers

.^[1]

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

Definition and properties[edit]

Let p be a prime number.

Integers[edit]

The p-adic valuation of an integer

is defined to be

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

where

denotes the set of natural numbers and

denotes divisibility of

by

. In particular,

is a function

.^[2]

For example,

,

, and

since

.

The notation

is sometimes used to mean

.^[3]

If

is a positive integer, then

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

;

this follows directly from

.

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,

and

since

.

Some properties are:

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

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

Moreover, if

, then

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

where

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

p-adic absolute value[edit]

The p-adic absolute value on

is the function

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

defined by

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

Thereby,

for all

and
for example,

and

The p-adic absolute value satisfies the following properties.

Non-negativity	${displaystyle |r|_{p}geq 0}$
Positive-definiteness	${displaystyle |r|_{p}=0iff r=0}$
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

it follows that

for the roots of unity

and

and consequently also

The subadditivity

follows from the non-Archimedean triangle inequality

.

The choice of base p in the exponentiation

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

. This follows from simply taking the prime factorization: each prime power factor

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

${displaystyle dcolon mathbb {Q} times mathbb {Q} to mathbb {R} _{geq 0}}$

defined by

${displaystyle d(r,s)=|r-s|_{p}.}$

The completion of

with respect to this metric leads to the set

of p-adic numbers.

See also[edit]

References[edit]