p-adic absolute value

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English

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Noun

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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 pk(a/b) — where a, b and k are integers and a, b and p are coprime — is mapped to the rational number p-k and 0 is mapped to 0. (Note: any nonzero rational number 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
    • 1993, Seth Warner, Topological Rings[1], Elsevier (North-Holland), page 8:
      If , then (with the convention ) is a nonarchimedean absolute value, denoted and called the p-adic absolute value to base . If and and if , then for every . The p-adic topology on is the topology defined by the p-adic absolute values.
    • 1999, Jan-Hendrik Evertse, Hans Peter Schlickewei, “The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group”, in Kálmán Györy, Henryk Iwaniec, Jerzy Urbanowicz, editors, Number Theory in Progress, Walter de Gruyter, page 121:
      They both gave essentially the same proof, based on the Subspace Theorem (more precisely, Schlickewei's generalisation to p-adic absolute values and number fields [30] of the Subspace Theorem proved by Schmidt in 1972 [41]).
    • 2007, Anthony W. Knapp, Advanced Algebra[2], Springer (Birkhäuser), page 320:
      It [] can also be defined as the subset [of ] with because the p-adic absolute value takes no values between 1 and , and therefore is open.

Usage notes

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  • 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).

Synonyms

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See also

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References

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

Further reading

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