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Definition 2025

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 $p^{k}{\Big (}{a \over b}{\Big )}$ — where a, b, and p are coprime and a, b, and k are integers — is mapped to the rational number $p^{-k}$ , 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

A notation for the p-adic absolute value of rational number x is $|x|_{p}$ .
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).