p-adic Norm
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
www.almerja.com
الجزء والصفحة:
...
13-10-2020
1089
p-adic Norm
Any nonzero rational number 
can be represented by
 |
(1)
|
where 
is a prime number, 
and 
are integers not divisible by 
, and 
is a unique integer. The p-adic norm of 
is then defined by
 |
(2)
|
Also define the 
-adic value
 |
(3)
|
As an example, consider the fraction
 |
(4)
|
It has 
-adic absolute values given by
The 
-adic norm of a nonzero rational number 
can be computed in the Wolfram Language as follows.
PadicNorm[x_Integer, p_Integer?PrimeQ] :=
p^(-IntegerExponent[x, p])
PadicNorm[x_Rational, p_Integer?PrimeQ] :=
PadicNorm[Numerator[x], p] /
PadicNorm[Denominator[x], p]
The 
-adic norm satisfies the relations
1. 
for all 
,
2. 
iff 
,
3. 
for all 
and 
,
4. 
for all 
and 
(the triangle inequality), and
5. 
for all 
and 
(the strong triangle inequality).
In the above, relation 4 follows trivially from relation 5, but relations 4 and 5 are relevant in the more general valuation theory.
The p-adic norm is the basis for the algebra of p-adic numbers.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
