# p-adic-absolute-value

Noun

(*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.)^{ }- 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

- A notation for the
*p*-adic absolute value of rational number*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 this*p*-adic absolute value and extends it to apply to*p*-adic irrationals, which could well be mapped to real numbers in general (not merely rationals).