p-adic absolute value

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English[edit]

Noun[edit]

p-adic absolute value (plural p-adic absolute values)

  1. (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

Usage notes[edit]

  • 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).

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References[edit]

  1. ^ 2008, Jacqui Ramagge, Unreal Numbers: The story of p-adic numbers (PDF file)