p-adic valuation

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

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] Let p be a prime number.

The p-adic valuation of an integer

denotes the set of natural numbers (including zero) and

denotes divisibility of

is a positive integer, then this follows directly from

The p-adic valuation can be extended to the rational numbers as the function defined by For example,

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

Legendre's formula shows that

For any positive integer n,

This infinite sum can be reduced to

This formula can be extended to negative integer values to give:

The p-adic absolute value (or p-adic norm,[6] though not a norm in the sense of analysis) on

is the function defined by Thereby,

The p-adic absolute value satisfies the following properties.

for the roots of unity

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: 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.

A metric space can be formed on the set

with a (non-Archimedean, translation-invariant) metric defined by The completion of

with respect to this metric leads to the set

of p-adic numbers.