p-adic

English

Etymology

From p +‎ -adic. The letter p follows common number theory usage in representing an arbitrary prime number; the suffix -adic signals that properties of the number p (generally, those due to its being prime, and specifically those of a specified p) are fundamental to the theory and determine the properties of the resulting construction.

Adjective

p-adic (not comparable)

(number theory) Of, pertaining to, (ultimately) derived from or defined in the context of p-adic numbers.
- 1995, Alain Escassut, Analytic Elements in P-adic Analysis, World Scientific, page v,
  The theory of p-adic analytic functions in domains other than simple disks is not very well known yet, although such kind of functions happens to intervene in questions linked to p-adic functional analysis, number theory, and others.
- 1997, Fernando Q. Gouvêa, p-adic Numbers: An Introduction, Springer, 2nd Edition, page 60,
  Definition 3.3.3 The ring of p-adic integers is the valuation ring
  ${\\displaystyle \\mathbb {Z} _{p}=\\{x\\in \\mathbb {Q} _{p}:|x|_{p}\\leq 1\\}}$ .
- 2007, Nguyen Minh Cheung, Nguyen Van Co, Le Quang Thuan, §12: Harmonic Analysis Over P-adic Field I: Some Equations and Singular Field Operators, N. M. Chuong, Yu V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (editors), Harmonic, Wavelet and P-adic Analysis, World Scientific, page 272,
  The paper is organized as follows:
  2. Preliminaries
  3. A p-adic Cauchy problem
  4. The p-adic Hilbert transform
  5. The boundaries in the p-adic space ${\\displaystyle L^{q}}$
  […]

Usage notes

The synonym ℓ-adic is used in some contexts: for instance, in regard to Galois representations (G-modules, where the Galois group G is a vector space over the field of ℓ-adic numbers), and in algebraic geometry, where ℓ-adic cohomology is one of the "classical" constructions of Weil cohomologies.

Synonyms

(of, etc., p-adic numbers): ℓ-adic

Derived terms

p-adic absolute value
p-adic norm
p-adic number
p-adic order
p-adic ordinal
p-adic ultrametric

Translations

Finnish: p-adinen
French: p-adique
German: p-adisch
Polish: not used in Polish
Romanian: p-adic