Any nonzero rational number can be represented by
(1)
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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)
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Also define the -adic value
(3)
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As an example, consider the fraction
(4)
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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)
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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.