''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 $ν$ such that $p ν$ divides $n$ . The $p$ -adic valuation of 0 is defined to be infinity. It is commonly denoted $ν p (n)$ . If $n / d$ is a rational number in lowest terms, so that $n$ and $d$ are relatively prime, then $ν p (n / d)$ is equal to $ν p (n)$ if $p$ divides $n$ , or $- ν p (d)$ 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 $ℤ$ . The $p$ -adic order or $p$ -adic valuation for $ℤ$ is defined as $ν p : ℤ \to ℕ$ ^[2]

\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 $\nu _{p}:\mathbb {Q} \to \mathbb {Z}$ ^[3]

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

Some properties are:

{\begin{aligned}\nu _{p}(m\cdot n)&=\nu _{p}(m)+\nu _{p}(n).\\[5px]\nu _{p}(m+n)&\geq \inf {\bigl \{}\nu _{p}(m),\nu _{p}(n){\bigr \}}.\end{aligned}}

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

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

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

p-adic absolute value

The $p$ -adic absolute value on $ℚ$ is defined as $| \cdot | p : ℚ \to ℝ$

|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:

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

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

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

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

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[1] Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. 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.