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 [math]\displaystyle{ \nu_p }[/math] such that [math]\displaystyle{ p^{\nu_p} }[/math] divides n. This function is easily extended to positive rational numbers r = a/b by
- [math]\displaystyle{ r = p_1^{\nu_{p_1}} p_2^{\nu_{p_2}} \cdots p_k^{\nu_{p_k}} = \prod_{i=1}^{k} p_i^{\nu_{p_i}} , }[/math]
where [math]\displaystyle{ p_1 \lt p_2 \lt \dotsb \lt p_k }[/math] are primes and the [math]\displaystyle{ \nu_{p_i} }[/math] are (unique) integers (considered to be 0 for all primes not occurring in r so that [math]\displaystyle{ \nu_{p_i}(r) = \nu_{p_i}(a) - \nu_{p_i}(b) }[/math]).
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 [math]\displaystyle{ \mathbb{Q} }[/math] p (relative to a chosen prime number p), whereas the completion with the usual absolute value results in the field of real numbers [math]\displaystyle{ \mathbb{R} }[/math].[1]
Definition and properties
Let p be a prime number.
Integers
The p-adic order or p-adic valuation for [math]\displaystyle{ \mathbb{Z} }[/math] is the function
- [math]\displaystyle{ \nu_p : \mathbb{Z} \to \mathbb{N} }[/math][2]
defined by
- [math]\displaystyle{ \nu_p(n)= \begin{cases} \mathrm{max}\{k \in \mathbb{N} : p^k \mid n\} & \text{if } n \neq 0\\ \infty & \text{if } n=0, \end{cases} }[/math]
where [math]\displaystyle{ \mathbb{N} }[/math] denotes the natural numbers.
For example, [math]\displaystyle{ \nu_3(-45) = 2 }[/math] and [math]\displaystyle{ \nu_5(-45) = 1 }[/math] since [math]\displaystyle{ |{-45}| = 45 = 3^2 \cdot 5^1 }[/math].
The notation [math]\displaystyle{ p^k \parallel n }[/math] is sometimes used to mean [math]\displaystyle{ k = \nu_p(n) }[/math].[3]
Rational numbers
The p-adic order can be extended into the rational numbers as the function
- [math]\displaystyle{ \nu_p : \mathbb{Q} \to \mathbb{Z} }[/math][4]
defined by
- [math]\displaystyle{ \nu_p\left(\frac{a}{b}\right)=\nu_p(a)-\nu_p(b). }[/math][5]
For example, [math]\displaystyle{ \nu_2 \bigl(\tfrac{9}{8}\bigr) = -3 }[/math] and [math]\displaystyle{ \nu_3 \bigl(\tfrac{9}{8}\bigr) = 2 }[/math] since [math]\displaystyle{ \tfrac{9}{8} = \tfrac{3^2}{2^3} }[/math].
Some properties are:
- [math]\displaystyle{ \begin{align} \nu_p(m\cdot n) &= \nu_p(m) + \nu_p(n) \\[5px] \nu_p(m+n) &\geq \min\bigl\{ \nu_p(m), \nu_p(n)\bigr\}. \end{align} }[/math]
Moreover, if [math]\displaystyle{ \nu_p(m) \neq \nu_p(n) }[/math], then
- [math]\displaystyle{ \nu_p(m+n)= \min\bigl\{ \nu_p(m), \nu_p(n)\bigr\} }[/math]
where min is the minimum (i.e. the smaller of the two).
p-adic absolute value
The p-adic absolute value on [math]\displaystyle{ \mathbb{Q} }[/math] is the function
- [math]\displaystyle{ |\cdot|_p \colon \Q \to \R_{\ge 0} }[/math]
defined by
- [math]\displaystyle{ |r|_p = p^{-\nu_p(r)} . }[/math][5]
For example, [math]\displaystyle{ |{-45}|_3 = \tfrac{1}{9} }[/math] and [math]\displaystyle{ \bigl|\tfrac{9}{8}\bigr|_2 = 8 . }[/math]
The p-adic absolute value satisfies the following properties.
Non-negativity [math]\displaystyle{ |a|_p \geq 0 }[/math] Positive-definiteness [math]\displaystyle{ |a|_p = 0 \iff a = 0 }[/math] Multiplicativity [math]\displaystyle{ |ab|_p = |a|_p|b|_p }[/math] Non-Archimedean [math]\displaystyle{ |a+b|_p \leq \max\left(|a|_p, |b|_p\right) }[/math]
The symmetry [math]\displaystyle{ |{-a}|_p = |a|_p }[/math] follows from multiplicativity [math]\displaystyle{ |ab|_p = |a|_p|b|_p }[/math] and the subadditivity [math]\displaystyle{ |a+b|_p \leq |a|_p + |b|_p }[/math] from the non-Archimedean triangle inequality [math]\displaystyle{ |a+b|_p \leq \max\left(|a|_p, |b|_p\right) }[/math].
The choice of base p in the exponentiation [math]\displaystyle{ p^{-\nu_p(r)} }[/math] makes no difference for most of the properties, but supports the product formula:
- [math]\displaystyle{ \prod_{0, p} |x|_p = 1 }[/math]
where the product is taken over all primes p and the usual absolute value, denoted [math]\displaystyle{ |x|_0 }[/math]. This follows from simply taking the prime factorization: each prime power factor [math]\displaystyle{ p^k }[/math] 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", 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 [math]\displaystyle{ \mathbb{Q} }[/math] with a (non-Archimedean, translation-invariant) metric
- [math]\displaystyle{ d \colon \Q \times \Q \to \R_{\ge 0} }[/math]
defined by
- [math]\displaystyle{ d(x,y) = |x-y|_p . }[/math]
The completion of [math]\displaystyle{ \mathbb{Q} }[/math] with respect to this metric leads to the field [math]\displaystyle{ \mathbb{Q} }[/math] p of p-adic numbers.
See also
References
- ↑ Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. pp. 758–759. 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]
- ↑ 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.
- ↑ Khrennikov, A.; Nilsson, M. (2004). p-adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.[ISBN missing]
- ↑ 5.0 5.1 with the usual rules for arithmetic operations
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