p-adic order

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.[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

Integers

Let p be a prime in Z. The p-adic order or p-adic valuation for Z is defined as[2]

Rational Numbers

The p-adic order can be extended into the rational numbers. We can define[3]

Some properties are:

Moreover, if , then

where is the Infimum (i.e. the smaller of the two)

p-adic Norm

From our definition of the p-adic order, we can define the so-called p-adic norm. (It is not actually a norm because it does not satisfy the requirement of homogeneity, but it is an absolute value.) 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

See also

References

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