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/ 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.
Definition and properties
where denotes the natural numbers.
For example, since .
p-adic absolute value
For example, and .
The p-adic absolute value satisfies the following properties.
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.
- Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
- Ireland, K.; Rosen, M. (2000). A Classical Introduction to Modern Number Theory. New York: Springer-Verlag. p. 3.
- Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.