p-adic order
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 extented 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 ℚ_{ p} (relative to a chosen prime number p), whereas the completion with the usual absolute value results in the field of real numbers ℝ.^{[1]}
Definition and propertiesEdit
Let p be a prime number.
IntegersEdit
The p-adic order or p-adic valuation for ℤ is the function
- ^{[2]}
defined by
where denotes the natural numbers.
For example, and since .
Rational numbersEdit
The p-adic order can be extended into the rational numbers as the function
- ^{[3]}
defined by
- ^{[4]}
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 valueEdit
The p-adic absolute value on ℚ is the function
defined by
- ^{[4]}
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 ℚ with a (non-Archimedean, translation-invariant) metric
defined by
The completion of ℚ with respect to this metric leads to the field ℚ_{ p} of p-adic numbers.
See alsoEdit
ReferencesEdit
- ^ Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
- ^ Ireland, K.; Rosen, M. (2000). A Classical Introduction to Modern Number Theory. New York: Springer-Verlag. p. 3.^{[ISBN missing]}
- ^ Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.^{[ISBN missing]}
- ^ ^{a} ^{b} with the usual rules for arithmetic operations