التقييم p -adic

التقييم p -adic لعدد صحيح${\displaystyle n}$يُعرَّف بأنه

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

أين${\displaystyle \mathbb {N} _{0}}$يرمز إلى مجموعة الأعداد الطبيعية (بما في ذلك الصفر) و${\displaystyle m\mid n}$يدل على قابلية القسمة${\displaystyle n}$بواسطة${\displaystyle m}$. بخاصة،${\displaystyle \nu _{p}}$هي دالة${\displaystyle \nu _{p}\colon \mathbb {Z} \to \mathbb {N} _{0}\cup \{\infty \}}$[ ^{2 ]​}

على سبيل المثال،${\displaystyle \nu _{2}(-12)=2}$،${\displaystyle \nu _{3}(-12)=1}$، و${\displaystyle \nu _{5}(-12)=0}$منذ${\displaystyle |{-12}|=12=2^{2}\cdot 3^{1}\cdot 5^{0}}$.

الترميز${\displaystyle p^{k}\parallel n}$يُستخدم أحيانًا بمعنى${\displaystyle k=\nu _{p}(n)}$[ ^{3 ]​}

لو${\displaystyle n}$إذا كان عددًا صحيحًا موجبًا، فإن${\displaystyle \nu _{p}(n)\leq \log _{p}n}$وهذا يتبع مباشرة من${\displaystyle n\geq p^{\nu _{p}(n)}}$.

يمكن توسيع التقييم p-adic ليشمل الأعداد النسبية كدالة

${\displaystyle \nu _{p}:\mathbb {Q} \to \mathbb {Z} \cup \{\infty \}}$^{[ 4 ] [ 5 ]}

محدد بواسطة

${\displaystyle \nu _{p}\left({\frac {r}{s}}\right)=\nu _{p}(r)-\nu _{p}(s).}$

على سبيل المثال،${\displaystyle \nu _{2}{\bigl (}{\tfrac {9}{8}}{\bigr )}=-3}$و${\displaystyle \nu _{3}{\bigl (}{\tfrac {9}{8}}{\bigr )}=2}$منذ${\displaystyle {\tfrac {9}{8}}=2^{-3}\cdot 3^{2}}$.

بعض الخصائص هي:

${\displaystyle \nu _{p}(r\cdot s)=\nu _{p}(r)+\nu _{p}(s)}$

${\displaystyle \nu _{p}(r+s)\geq \min {\bigl \{}\nu _{p}(r),\nu _{p}(s){\bigr \}}}$

علاوة على ذلك، إذا${\displaystyle \nu _{p}(r)\neq \nu _{p}(s)}$، ثم

${\displaystyle \nu _{p}(r+s)=\min {\bigl \{}\nu _{p}(r),\nu _{p}(s){\bigr \}}}$

أين${\displaystyle \min }$هو الحد الأدنى (أي الأصغر من الاثنين).

تُظهر معادلة ليجندر أن${\displaystyle \nu _{p}(n!)=\sum _{i=1}^{\infty {}}{\left\lfloor {\frac {n}{p^{i}}}\right\rfloor {}}}$.

لأي عدد صحيح موجب n ،${\displaystyle n={\frac {n!}{(n-1)!}}}$وهكذا${\displaystyle \nu _{p}(n)=\nu _{p}(n!)-\nu _{p}((n-1)!)}$.

لذلك،${\displaystyle \nu {}_{p}(n)=\sum _{i=1}^{\infty {}}{{\bigg (}\left\lfloor {\frac {n}{p^{i}}}\right\rfloor {}-\left\lfloor {\frac {n-1}{p^{i}}}\right\rfloor {}{\bigg )}}}$.

يمكن اختزال هذا المجموع اللانهائي إلى${\displaystyle \sum _{i=1}^{\lfloor {\log _{p}{(n)}\rfloor {}}}{{\bigg (}\left\lfloor {\frac {n}{p^{i}}}\right\rfloor {}-\left\lfloor {\frac {n-1}{p^{i}}}\right\rfloor {}{\bigg )}}}$.

يمكن توسيع هذه الصيغة لتشمل قيم الأعداد الصحيحة السالبة، لنحصل على:

${\displaystyle \nu {}_{p}(n)=\sum _{i=1}^{\lfloor {\log _{p}{(|n|)}\rfloor {}}}{{\bigg (}\left\lfloor {\frac {|n|}{p^{i}}}\right\rfloor {}-\left\lfloor {\frac {|n|-1}{p^{i}}}\right\rfloor {}{\bigg )}}}$

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

${\displaystyle \prod _{0,p}|r|_{p}=1}$

where the product is taken over all primes p and the usual absolute value, denoted ${\displaystyle |r|_{0}}$. This follows from simply taking the prime factorization: each prime power factor ${\displaystyle p^{k}}$ contributes its reciprocal to its p-adic absolute value, and then the usual Archimedean absolute value cancels all of them.

By Ostrowski's theorem, the usual and p-adic absolute values occurring in the formula are all the absolute values on the rational numbers up to equivalence. An analogous product formula can be used to axiomatically define global fields, of which the rational numbers are the simplest example.

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

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

defined by

${\displaystyle d(r,s)=|r-s|_{p}.}$

The completion of ${\displaystyle \mathbb {Q} }$ with respect to this metric leads to the set ${\displaystyle \mathbb {Q} _{p}}$ of p-adic numbers. Like the rationals, they form a field, and the p-adic valuation and absolute value can be extended to ${\displaystyle \mathbb {Q} _{p}}$, making it a complete valued field.