p-adic order

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In basic number theory, for a given prime number p, the p-adic order of a positive integer n is the highest exponent such that divides n. This function is easily extended to positive rational numbers r = a/b by

where are primes and the are (unique) integers (considered to be 0 for all primes not occurring in r so that ).

This p-adic order constitutes an (additively written) valuation, the so-called p-adic valuation, which when written multiplicatively is an analogue to the well-known usual absolute value. Both types of valuations can be used for completing the field of rational numbers, where the completion with a p-adic valuation results in a field of p-adic numbers p (relative to a chosen prime number p), whereas the completion with the usual absolute value results in the field of real 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.

Integers[edit]

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

[2]

defined by

where denotes the natural numbers.

For example, and since .

The notation is sometimes used to mean .[3]

Rational numbers[edit]

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

[4]

defined by

[5]

For example, and since .

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 the function

defined by

[5]

For example, and

The p-adic absolute value satisfies the following properties.

Non-negativity
Positive-definiteness
Multiplicativity
Non-Archimedean

The symmetry follows from multiplicativity and the subadditivity from the non-Archimedean triangle inequality .

The choice of base p in the exponentiation makes no difference for most of the properties, but supports the product formula:

where the product is taken over all primes p and the usual absolute value, 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.

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

A metric space can be formed on the set with a (non-Archimedean, translation-invariant) metric

defined by

The completion of with respect to this metric leads to the field p of p-adic numbers.

See also[edit]

References[edit]

  1. ^ Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. pp. 758–759. 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. ^ Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons. p. 4. ISBN 0-471-62546-9.
  4. ^ Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.[ISBN missing]
  5. ^ a b with the usual rules for arithmetic operations