''p''-adic order

In number theory, for a given prime number $p$ , the $p$ -adic order or $p$ -adic valuation of a non-zero integer $n$ is the highest exponent $\nu$ such that $p^{\nu }$ divides $n$ . The $p$ -adic valuation of $0$ is defined to be $\infty$ . It is commonly denoted $\nu _{p}(n)$ . If $n/d$ is a rational number in lowest terms, so that $n$ and $d$ are relatively prime, then $\nu _{p}(n/d)$ is equal to $\nu _{p}(n)$ if $p$ divides $n$ , or $-\nu _{p}(d)$ if $p$ divides $d$ , or to $0$ if it divides neither. The most important application of the $p$ -adic order is in constructing the field of $p$ -adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even 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

Integers

Let $p$ be a prime in $\mathbb {Z}$ . The $p$ -adic order or $p$ -adic valuation for $\mathbb {Z}$ is defined as^[2] $\nu _{p}:\mathbb {Z} \to \mathbb {N}$

\nu _{p}(n)={\begin{cases}\mathrm {max} \{v\in \mathbb {N} :p^{v}\mid n\}&{\text{if }}n\neq 0\\\infty &{\text{if }}n=0\end{cases}}

Rational numbers

The $p$ -adic order can be extended into the rational numbers. We can define^[3] $\nu _{p}:\mathbb {Q} \to \mathbb {Z}$

\nu_p\left(\frac{a}{b}\right)=\nu_p(a)-\nu_p(b).

Some properties are:

\nu_p(m\cdot n)= \nu_p(m) + \nu_p(n)~.

\nu_p(m+n)\geq \inf\{ \nu_p(m), \nu_p(n)\}.

Moreover, if

\nu_p(m)\ne \nu_p(n)

, then

\nu_p(m+n)= \inf\{ \nu_p(m), \nu_p(n)\}.

where $\inf$ is the infimum (i.e. the smaller of the two).

$p$ -adic absolute value

The $p$ -adic absolute value on $\mathbb {Q}$ is defined as $|\,\cdot \,|_{p}:\mathbb {Q} \to \mathbb {R}$

|x|_p = \begin{cases} p^{-\nu_p(x)} & \text{if } x \neq 0\\ 0 & \text{if } x=0 \end{cases}

The $p$ -adic absolute value satisfies the following properties.

\begin{align} |a|_p \geq 0 & \quad \text{Non-negativity}\\ |a|_p = 0 \iff a = 0 & \quad \text{Positive-definiteness}\\ |ab|_p = |a|_p|b|_p & \quad \text{Multiplicativeness}\\ |a+b|_p \leq |a|_p + |b|_p & \quad \text{Subadditivity}\\ |a+b|_p \leq \max\left(|a|_p, |b|_p\right) & \quad \text{it is non-archimedean}\\ |-a|_p = |a|_p & \quad \text{Symmetry} \end{align}

A metric space can be formed on the set $\mathbb {Q}$ with a (non-archimedean, translation invariant) metric defined by $d:\mathbb {Q} \times \mathbb {Q} \to \mathbb {R}$

d(x,y)=|x-y|_p.

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.

References

↑ David S. Dummit; Richard M. Foote (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 0-471-43334-9.
↑ Ireland, K., Rosen, M. (2000). A Classical Introduction to Modern Number Theory. Springer-Verlag New York. Inc., p. 3
↑ Khrennikov, A., Nilsson, M. (2004). P-adic Deterministic and Random Dynamics., Kluwer Academic Publishers, p. 9

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