$p$ -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 $\nu _{p}$ such that $p^{\nu _{p}}$ divides $n$ . This function is easily extented to positive rational numbers $r = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} a / b$ by

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}}},

where $p_{1}<p_{2}<\dotsb <p_{k}$ are primes and the $\nu _{p_{i}}$ are (unique) integers (considered to be 0 for all primes not occurring in $r$ so that $\nu _{p_{i}}(r)=\nu _{p_{i}}(a)-\nu _{p_{i}}(b)$ ).

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]

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

Let $p$ be a prime number.

Integers

The $p$ -adic order or $p$ -adic valuation for $ℤ$ is the function

\nu _{p}:\mathbb {Z} \to \mathbb {N}

[2]

defined by

\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}}

where $\mathbb {N}$ denotes the natural numbers.

For example, $\nu _{3}(-45)=2$ and $\nu _{5}(-45)=1$ since $|{-45}|=45=3^{2}\cdot 5^{1}$ .

Rational numbers

The $p$ -adic order can be extended into the rational numbers as the function

\nu _{p}:\mathbb {Q} \to \mathbb {Z}

[3]

defined by

\nu _{p}\left({\frac {a}{b}}\right)=\nu _{p}(a)-\nu _{p}(b).

[4]

For example, $\nu _{2}{\bigl (}{\tfrac {9}{8}}{\bigr )}=-3$ and $\nu _{3}{\bigl (}{\tfrac {9}{8}}{\bigr )}=2$ since ${\tfrac {9}{8}}={\tfrac {3^{2}}{2^{3}}}$ .

Some properties are:

{\begin{aligned}\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{aligned}}

Moreover, if $\nu _{p}(m)\neq \nu _{p}(n)$ , then

\nu _{p}(m+n)=\min {\bigl \{}\nu _{p}(m),\nu _{p}(n){\bigr \}}

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

$p$ -adic absolute value

The $p$ -adic absolute value on $ℚ$ is the function

|\cdot |_{p}\colon \mathbb {Q} \to \mathbb {R} _{\geq 0}

defined by

|r|_{p}=p^{-\nu _{p}(r)}.

[4]

For example, $|{-45}|_{3}={\tfrac {1}{9}}$ and ${\bigl |}{\tfrac {9}{8}}{\bigr |}_{2}=8.$

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

Non-negativity	$\|a\|_{p}\geq 0$
Positive-definiteness	$\|a\|_{p}=0\iff a=0$
Multiplicativity	$\|ab\|_{p}=\|a\|_{p}\|b\|_{p}$
Non-Archimedean	$\|a+b\|_{p}\leq \max \left(\|a\|_{p},\|b\|_{p}\right)$

The symmetry $|{-a}|_{p}=|a|_{p}$ follows from multiplicativity $|ab|_{p}=|a|_{p}|b|_{p}$ and the subadditivity $|a+b|_{p}\leq |a|_{p}+|b|_{p}$ from the non-Archimedean triangle inequality $|a+b|_{p}\leq \max \left(|a|_{p},|b|_{p}\right)$ .

The choice of base $p$ in the exponentiation $p^{-\nu _{p}(r)}$ makes no difference for most of the properties, but supports the product formula:

\prod _{0,p}|x|_{p}=1

where the product is taken over all primes $p$ and the usual absolute value, denoted $|x|_{0}$ . This follows from simply taking the prime factorization: each prime power factor $p^{k}$ 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 $ℚ$ with a (non-Archimedean, translation-invariant) metric

d\colon \mathbb {Q} \times \mathbb {Q} \to \mathbb {R} _{\geq 0}

defined by

d(x,y)=|x-y|_{p}.

The completion of $ℚ$ with respect to this metric leads to the field $ℚ p$ of $p$ -adic numbers.

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.
Khrennikov, A.; Nilsson, M. (2004). $p$ -adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.
with the usual rules for arithmetic operations

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[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.

[3] Khrennikov, A.; Nilsson, M. (2004). $p$ -adic Deterministic and Random Dynamics. Kluwer Academic Publishers. p. 9.

[infty-4] with the usual rules for arithmetic operations

p-adic order

Definition and properties

Integers

Rational numbers

p-adic absolute value

See also

References

$p$ -adic order

$p$ -adic absolute value