In number theory, for a given prime number p, the p-adic order or p-adic additive valuation of a non-zero integer n is the highest exponent ν such that pν divides n. The p-adic valuation of is defined to be . It is commonly abbreviated νp(n). If n/d is a rational number in lowest terms, so that n and d are relatively prime, then νp(n/d) is equal to νp(n) if p divides n, or -νp(d) if p divides d, or to 0 if it divides neither one. 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
Let p be a prime in Z. The p-adic order or p-adic valuation for Z is defined as
The p-adic order can be extended into the rational numbers. We can define
Some properties are:
- Moreover, if , then
where is the Infimum (i.e. the smaller of the two)
From our definition of the p-adic order, we can define the p-adic norm. The p-adic norm of Q is defined as
Some properties of the p-adic norm:
A metric space can be formed on the set Q with a (non-archimedean, translation invariant) metric defined by
- David S. Dummit; Richard M. Foote (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
- Ireland, K., Rosen, M. (2000). A Classical Introduction to Modern Number Theory. Springer-Verlag New York. Inc., p. 3
- Khrennikov, A., Nilsson, M. (2004). P-adic Deterministic and Random Dynamics., Kluwer Academic Publishers, p. 9