p-adic absolute value

English

Noun

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(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.)

   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, Elsevier (North-Holland), page 8:

     If ${\displaystyle c>1}$, then ${\displaystyle x\rightarrow c^{-v_{p}(x)}}$ (with the convention ${\displaystyle c^{-\infty }=0}$) is a nonarchimedean absolute value, denoted ${\displaystyle \vert ..\vert _{p,c}}$ and called the p-adic absolute value to base ${\displaystyle c}$. If ${\displaystyle c>1}$ and ${\displaystyle d>1}$ and if ${\displaystyle r=\log _{c}d}$, then ${\displaystyle \vert x\vert _{p,d}=\vert x\vert _{p,c}^{r}}$ for every ${\displaystyle x\in K}$. The p-adic topology on ${\displaystyle K}$ 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, Springer (Birkhäuser), page 320:

     It ^{[${\displaystyle \mathbb {Z} _{p}}$]} can also be defined as the subset ^{[of ${\displaystyle \mathbb {Q} _{p}}$]} with ${\displaystyle \vert x\vert _{p}<p}$ because the p-adic absolute value takes no values between 1 and ${\displaystyle p}$, and therefore ${\displaystyle \mathbb {Z} _{p}}$ is open.

Usage notes

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

Synonyms

• p-adic norm

Hypernyms

• norm

See also

• p-adic order, p-adic valuation
• p-adic ultrametric

References

Further reading

• p-adic order on Wikipedia.Wikipedia
• Absolute value (algebra) on Wikipedia.Wikipedia