P-adic order


In number theory, for a given prime number, the -adic order or -adic valuation of a non-zero integer is the highest exponent such that divides. The -adic valuation of 0 is defined to be infinity. The -adic valuation is commonly denoted. If is a rational number in lowest terms, so that and are coprime, then is equal to if divides, or if divides, or to 0 if it divides neither. The most important application of the -adic order is in constructing the field of -adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.

Definition and properties

Integers

Let be a prime number. The -adic order or -adic valuation for is the function defined by
where denotes the natural numbers.
For example, since.

Rational numbers

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

''p''-adic absolute value

The -adic absolute value on is defined as
For example, and.
The -adic absolute value satisfies the following properties.
The symmetry follows from multiplicativity and subadditivity from the non-Archimedean triangle inequality.
A metric space can be formed on the set with a metric defined by
The -adic absolute value is sometimes referred to as the "-adic norm", although it is not actually a norm because it does not satisfy the requirement of homogeneity.
The choice of base in the formula makes no difference for most of the properties, but results in 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.