The p-adic absolute value (or p-adic norm,[citation needed] though not a norm in the sense of analysis) on is the function
defined by
Thereby, for all and
for example, and
The p-adic absolute value satisfies the following properties.
Non-negativity | |
Positive-definiteness | |
Multiplicativity | |
Non-Archimedean | |
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:
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.