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

			(5)
			(6)
			(7)
			(8)
			(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.