P-adic valuation

In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides.
It is denoted.
Equivalently, is the exponent to which appears in the prime factorization of.
The -adic valuation is a valuation and gives rise to an analogue of the usual absolute value.
Whereas the Complete [metric space|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 numbers.
[Image:2adic12480.svg|thumb|right|200px|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

Let be a prime number.

Integers

The -adic valuation of an integer is defined to be
where denotes the set of natural numbers and denotes divisibility of by. In particular, is a function.
For example,,, and since.
The notation is sometimes used to mean.
If is a positive integer, then
this follows directly from.

Rational numbers

The -adic valuation can be extended to the rational numbers as the function
defined by
For example, and since.
Some properties are:
Moreover, if, then
where is the minimum.

Formula for the -adic valuation of Integers

Legendre's formula shows that.
For any positive integer, and so.
Therefore,.
This infinite sum can be reduced to.
This formula can be extended to negative integer values to give:

-adic absolute value

The -adic absolute value on [rational number|] is the function
defined by
Thereby, for all and
for example, and
The -adic absolute value satisfies the following properties.
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 in the exponentiation makes no difference for most of the properties, but supports the product formula:
where the product is taken over all primes and the usual absolute value, denoted. This follows from simply taking the prime factorization: each prime power factor contributes its reciprocal to its -adic absolute value, and then the usual Archimedean absolute value cancels all of them.
A metric space can be formed on the set with a metric
defined by
The completion of with respect to this metric leads to the set of -adic numbers.