P-adic valuation#p-adic absolute value

Let {{mvar|p}} be a prime number.

The {{mvar|p}}-adic valuation of an integer $n$ is defined to be

:$$

\nu_p(n)=

\begin{cases}

\mathrm{max}\{k \in \mathbb{N}_0 : p^k \mid n\} & \text{if } n \neq 0\\

\infty & \text{if } n=0,

\end{cases}

where $\backslash mathbb\{N\}\_0$ denotes the set of natural numbers (including zero) and $m\; \backslash mid\; n$ denotes divisibility of $n$ by $m$. In particular, $\backslash nu\_p$ is a function $\backslash nu\_p\; \backslash colon\; \backslash mathbb\{Z\}\; \backslash to\; \backslash mathbb\{N\}\_0\; \backslash cup\backslash \{\backslash infty\backslash \}$.{{cite book|last1=Ireland |first1=K. |last2=Rosen |first2=M. |date=2000 |title=A Classical Introduction to Modern Number Theory |publisher=Springer-Verlag |location=New York |page=3}}{{ISBN needed}}

For example, $\backslash nu\_2(-12)\; =\; 2$, $\backslash nu\_3(-12)\; =\; 1$, and $\backslash nu\_5(-12)\; =\; 0$ since $|\{-12\}|\; =\; 12\; =\; 2^2\; \backslash cdot\; 3^1\; \backslash cdot\; 5^0$.

The notation $p^k\; \backslash parallel\; n$ is sometimes used to mean $k\; =\; \backslash nu\_p(n)$.{{Cite book |last1=Niven |first1=Ivan |author1-link=Ivan M. Niven |last2=Zuckerman |first2=Herbert S. |last3=Montgomery |first3=Hugh L. |author3-link=Hugh Lowell Montgomery |title=An Introduction to the Theory of Numbers |date=1991 |publisher=John Wiley & Sons |edition=5th |isbn=0-471-62546-9 |page=4}}

If $n$ is a positive integer, then

:$\backslash nu\_p(n)\; \backslash leq\; \backslash log\_p\; n$;

this follows directly from $n\; \backslash geq\; p^\{\backslash nu\_p(n)\}$.

The {{mvar|p}}-adic valuation can be extended to the rational numbers as the function

:$\backslash nu\_p\; :\; \backslash mathbb\{Q\}\; \backslash to\; \backslash mathbb\{Z\}\; \backslash cup\backslash \{\backslash infty\backslash \}$with the usual order relation, namely

:$\backslash infty\; >\; n$,

and rules for arithmetic operations,

:$\backslash infty\; +\; n\; =\; n\; +\; \backslash infty\; =\; \backslash infty$,

on the extended number line.{{cite book|last1=Khrennikov |first1=A. |last2=Nilsson |first2=M. |date=2004 |title={{mvar|p}}-adic Deterministic and Random Dynamics |publisher=Kluwer Academic Publishers |page=9}}{{ISBN needed}}

defined by

:$$

\nu_p\left(\frac{r}{s}\right)=\nu_p(r)-\nu_p(s).

For example, $\backslash nu\_2\; \backslash bigl(\backslash tfrac\{9\}\{8\}\backslash bigr)\; =\; -3$ and $\backslash nu\_3\; \backslash bigl(\backslash tfrac\{9\}\{8\}\backslash bigr)\; =\; 2$ since $\backslash tfrac\{9\}\{8\}\; =\; 2^\{-3\}\backslash cdot\; 3^2$.

Some properties are:

:$\backslash nu\_p(r\backslash cdot\; s)\; =\; \backslash nu\_p(r)\; +\; \backslash nu\_p(s)$

:$\backslash nu\_p(r+s)\; \backslash geq\; \backslash min\backslash bigl\backslash \{\; \backslash nu\_p(r),\; \backslash nu\_p(s)\backslash bigr\backslash \}$

Moreover, if $\backslash nu\_p(r)\; \backslash neq\; \backslash nu\_p(s)$, then

:$\backslash nu\_p(r+s)=\; \backslash min\backslash bigl\backslash \{\; \backslash nu\_p(r),\; \backslash nu\_p(s)\backslash bigr\backslash \}$

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

Legendre's formula shows that $\backslash nu\_p(n!)=\backslash sum\_\{i=1\}^\{\backslash infty\{\}\}\{\backslash left\backslash lfloor\{\backslash frac\{n\}\{p^i\}\}\backslash right\backslash rfloor\{\}\}$.

For any positive integer {{mvar|n}}, $n\; =\; \backslash frac\{n!\}\{(n-1)!\}$ and so $\backslash nu\_p(n)=\backslash nu\_p(n!)-\backslash nu\_p((n-1)!)$.

Therefore, $\backslash nu\{\}\_p(n)=\backslash sum\_\{i=1\}^\{\backslash infty\{\}\}\{\backslash bigg(\backslash left\backslash lfloor\{\backslash frac\{n\}\{p^i\}\}\backslash right\backslash rfloor\{\}-\backslash left\backslash lfloor\{\backslash frac\{n-1\}\{p^i\}\}\backslash right\backslash rfloor\{\}\backslash bigg)\}$.

This infinite sum can be reduced to $\backslash sum\_\{i=1\}^\{\backslash lfloor\{\backslash log\_p\{(n)\}\backslash rfloor\{\}\}\}\{\backslash bigg(\backslash left\backslash lfloor\{\backslash frac\{n\}\{p^i\}\}\backslash right\backslash rfloor\{\}-\backslash left\backslash lfloor\{\backslash frac\{n-1\}\{p^i\}\}\backslash right\backslash rfloor\{\}\backslash bigg)\}$.

This formula can be extended to negative integer values to give:

$\backslash nu\{\}\_p(n)\; =\backslash sum\_\{i=1\}^\{\backslash lfloor\{\backslash log\_p\{(|n|)\}\backslash rfloor\{\}\}\}\{\backslash bigg(\backslash left\backslash lfloor\{\backslash frac$

{{anchor|p-adic norm}}

The {{mvar|p}}-adic absolute value (or {{mvar|p}}-adic norm,{{cite book

| last = Murty | first = M. Ram

| doi = 10.1007/978-1-4757-3441-6

| isbn = 0-387-95143-1

| mr = 1803093

| pages = 147–148

| publisher = Springer-Verlag, New York

| series = Graduate Texts in Mathematics

| title = Problems in analytic number theory

| volume = 206

| year = 2001}} though not a norm in the sense of analysis) on $\backslash mathbb\{Q\}$ is the function

:$|\backslash cdot|\_p\; \backslash colon\; \backslash Q\; \backslash to\; \backslash R\_\{\backslash ge\; 0\}$

defined by

:$|r|\_p\; =\; p^\{-\backslash nu\_p(r)\}\; .$

Thereby, $|0|\_p\; =\; p^\{-\backslash infty\}\; =\; 0$ for all $p$ and

for example, $|\{-12\}|\_2\; =\; 2^\{-2\}\; =\; \backslash tfrac\{1\}\{4\}$ and $\backslash bigl|\backslash tfrac\{9\}\{8\}\backslash bigr|\_2\; =\; 2^\{-(-3)\}\; =\; 8\; .$

The {{mvar|p}}-adic absolute value satisfies the following properties.

:

From the multiplicativity $|r\; s|\_p\; =\; |r|\_p|s|\_p$ it follows that $|1|\_p=1=|-1|\_p$ for the roots of unity $1$ and $-1$ and consequently also $|\{-r\}|\_p\; =\; |r|\_p\; .$

The subadditivity $|r+s|\_p\; \backslash leq\; |r|\_p\; +\; |s|\_p$ follows from the non-Archimedean triangle inequality $|r+s|\_p\; \backslash leq\; \backslash max\backslash left(|r|\_p,\; |s|\_p\backslash right)$.

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

:$\backslash prod\_\{0,\; p\}\; |r|\_p\; =\; 1$

where the product is taken over all primes {{mvar|p}} and the usual absolute value, denoted $|r|\_0$. This follows from simply taking the prime factorization: each prime power factor $p^k$ contributes its reciprocal to its {{mvar|p}}-adic absolute value, and then the usual Archimedean absolute value cancels all of them.

A metric space can be formed on the set $\backslash mathbb\{Q\}$ with a (non-Archimedean, translation-invariant) metric

:$d\; \backslash colon\; \backslash Q\; \backslash times\; \backslash Q\; \backslash to\; \backslash R\_\{\backslash ge\; 0\}$

defined by

:$d(r,s)\; =\; |r-s|\_p\; .$

The completion of $\backslash mathbb\{Q\}$ with respect to this metric leads to the set $\backslash mathbb\{Q\}\_p$ of {{mvar|p}}-adic numbers.

• P-adic number
• Valuation (algebra)
• Archimedean property
• Multiplicity (mathematics)
• Ostrowski's theorem
• Legendre's formula, for the $p$-adic valuation of $n!$
• Lifting-the-exponent lemma, for the $p$-adic valuation of $a^n-b^n$

{{reflist}}

Category:Algebraic number theory

Category:p-adic numbers