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.