TOPICS
Search

p-adic Norm


Any nonzero rational number x can be represented by

 x=(p^ar)/s,
(1)

where p is a prime number, r and s are integers not divisible by p, and a is a unique integer. The p-adic norm of x is then defined by

 |x|_p=p^(-a).
(2)

Also define the p-adic value

 |0|_p=0.
(3)

As an example, consider the fraction

 (140)/(297)=2^2·3^(-3)·5·7·11^(-1).
(4)

It has p-adic absolute values given by

|(140)/(297)|_2=1/4
(5)
|(140)/(297)|_3=27
(6)
|(140)/(297)|_5=1/5
(7)
|(140)/(297)|_7=1/7
(8)
|(140)/(297)|_(11)=11.
(9)

The p-adic norm of a nonzero rational number x 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 p-adic norm satisfies the relations

1. |x|_p>=0 for all x,

2. |x|_p=0 iff x=0,

3. |xy|_p=|x|_p|y|_p for all x and y,

4. |x+y|_p<=|x|_p+|y|_p for all x and y (the triangle inequality), and

5. |x+y|_p<=max(|x|_p,|y|_p) for all x and y (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.


See also

Greatest Dividing Exponent, p-adic Number

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "p-adic Norm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/p-adicNorm.html

Subject classifications