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Date: 22-10-2019
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Date: 18-7-2020
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Date: 6-2-2020
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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
(5) |
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(6) |
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(7) |
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(8) |
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(9) |
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.
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