p-adic absolute value

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.) [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, 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, 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

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

Further reading

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