p-adic valuation – Wikipedia
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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]
Definition and properties[edit]
Let p be a prime number.
Integers[edit]
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[edit]
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[edit]
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[edit]
References[edit]
- ^
- ^ Ireland, K.; Rosen, M. (2000). A Classical Introduction to Modern Number Theory. New York: Springer-Verlag. p. 3.[ISBN missing]
- ^ 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.
- ^ with the usual order relation, namely
- ,
on the extended number line.
- ^ Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.[ISBN missing]
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