p-adic order

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In number theory, for a given prime number p, the p-adic order or p-adic valuation of a non-zero integer n is the highest exponent such that divides n. The p-adic valuation of 0 is defined to be infinity. The p-adic valuation is commonly denoted .

If n/d is a rational number in lowest terms, so that n and d are coprime, then is equal to if p divides n, or if p divides d, or to 0 if it divides neither.

The most important application of the p-adic order is in constructing the field of p-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.[1]

Distribution of natural numbers by their 2-adic order, labeled with corresponding powers of two in decimal. Zero always has an infinite order

Definition and properties[edit]

Let p be a prime number.


The p-adic order or p-adic valuation for is the function


defined by

where denotes the natural numbers.

For example,

since .

Rational numbers[edit]

The p-adic order can be extended into the rational numbers as the function


defined by

For example,


Some properties are:

Moreover, if , then

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

p-adic absolute value[edit]

The p-adic absolute value on is defined as

|·|p :

For example,



The p-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 (non-Archimedean, translation-invariant) metric defined by d : ×

The p-adic absolute value is sometimes referred to as the "p-adic norm", although it is not actually a norm because it does not satisfy the requirement of homogeneity.

The choice of base p 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 p and the usual absolute value (Archimedean norm), 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.

See also[edit]


  1. ^ Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
  2. ^ Ireland, K.; Rosen, M. (2000). A Classical Introduction to Modern Number Theory. New York: Springer-Verlag. p. 3.[ISBN missing]
  3. ^ Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.[ISBN missing]