Wikipedia / P-adic valuation
Wikipedia / 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 .
The p-adicadic valuation is a valuation and gives rise to an analogue of the usual absolute value, though unlike the latter, the -adic absolute value is not Archimedean. 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.
Definition and properties
Let be a prime number.
Integers
The -adic valuation is defined to be
denotes the set of natural numbers denotes divisibility.
.
.
.
Rational numbers
The -adic valuation can be extended to the rational numbers as the function
defined by
.
Some properties are:
, then
is the minimum (i.e. the smaller of the two).
Formula for the p-adic valuation of integers
For any positive integer .
.
.
This formula can be extended to negative integer values to give:
p-adic absolute value
The -adic absolute value (or -adic norm, though not a norm in the sense of analysis) on \mathbb{Q} is the function
defined by
The -adic absolute value satisfies the following properties.
{| class="wikitable"
|- |Positive-definiteness |- |Multiplicativity |- |Non-Archimedean |} From the multiplicativity for the roots of unity The subadditivity follows from the non-Archimedean triangle inequality.
Product formula
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 . This follows from simply taking the prime factorization contributes its reciprocal to its -adic absolute value, and then the usual Archimedean absolute value cancels all of them.
By Ostrowski's theorem, the usual and -adic absolute values occurring in the formula are all the absolute values on the rational numbers up to equivalence. An analogous product formula can be used to axiomatically define global fields, of which the rational numbers are the simplest example.
Metric and completion
A metric space with a (non-Archimedean, translation-invariant) metric
defined by
The completion of -adic numbers. Like the rationals, they form a field, and the , making it a complete valued field.
See also
References