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 ν such that pν divides n. The p-adic valuation of 0 is defined to be infinity. It is commonly denoted 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. 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]

Integers[edit]

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

Rational numbers[edit]

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

Some properties are:

Moreover, if νp(m) ≠ νp(n), then

where inf is the infimum (i.e. the smaller of the two).

p-adic absolute value[edit]

The p-adic absolute value on is defined as |·|p :

The p-adic absolute value satisfies the following properties:

Non-negativity
Positive-definiteness
Multiplicativity
Subadditivity
Non-Archimedean[disambiguation needed]
Symmetry

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.

See also[edit]

References[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]