In number theory, the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted . Equivalently, is the exponent to which appears in the prime factorization of .

The p-adic valuation is a valuation and gives rise to an analogue of the usual absolute value. Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers , the completion of the rational numbers with respect to the -adic absolute value results in the p-adic numbers .[1]

Distribution of natural numbers by their 2-adic valuation, labeled with corresponding powers of two in decimal. Zero has an infinite valuation.
Distribution of natural numbers by their 2-adic valuation, labeled with corresponding powers of two in decimal. Zero has an infinite valuation.

Definition and properties

Let p be a prime number.

Integers

The p-adic valuation of an integer is defined to be

where denotes the set of natural numbers and denotes divisibility of by . In particular, is a function .[2]

For example, , , and since .

The notation is sometimes used to mean .[3]

If is a positive integer, then

;

this follows directly from .

Rational numbers

The p-adic valuation can be extended to the rational numbers as the function

[4][5]

defined by

For example, and since .

Some properties are:

Moreover, if , then

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

p-adic absolute value

The p-adic absolute value on is the function

defined by

Thereby, for all and for example, and

The p-adic absolute value satisfies the following properties.

Non-negativity
Positive-definiteness
Multiplicativity
Non-Archimedean

From the multiplicativity it follows that for the roots of unity and and consequently also The subadditivity follows 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 set of p-adic numbers.

See also

References

  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. ^ with the usual order relation, namely
    ,
    and rules for arithmetic operations,
    ,
    on the extended number line.
  5. ^ Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.[ISBN missing]