# p-adic absolute value

## Contents

## English[edit]

### Noun[edit]

** p-adic absolute value** (

*plural*

**p-adic absolute values**)

- (number theory, field theory) a norm for the rational numbers, with some prime number
*p*as parameter, such that any rational number of the form — where*a*,*b*, and*p*are coprime and*a*,*b*, and*k*are integers — is mapped to the rational number , and 0 is mapped to 0. (*Note:*any rational number, except 0, can be reduced to such a form.)^{[1]}- According to Ostrowski's theorem, only three kinds of norms are possible for the set of real numbers: the trivial absolute value, the real absolute value, and the
.*p*-adic absolute value^{WP}

- According to Ostrowski's theorem, only three kinds of norms are possible for the set of real numbers: the trivial absolute value, the real absolute value, and the

#### Usage notes[edit]

- A notation for the
of rational number*p*-adic absolute value*x*is . - The function is actually from the set of rational numbers to the set of real numbers, because it is used to construct/define a completion of the set of real numbers, namely, the field of
*p*-adic numbers, and this field inherits thisand extends it to apply to*p*-adic absolute value*p*-adic irrationals, which could well be mapped to real numbers in general (not merely rationals).

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### References[edit]

- ^
**2008**, Jacqui Ramagge, Unreal Numbers: The story of*p*-adic numbers (PDF file)