DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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The p-adic metric on Zp is determined by the first coordinate at which two compatible residue systems differ

Definition

For $x = (x_{n})$ and $y = (y_{n})$ in $Z_{p}$ , let

$ν_{p} (x, y) : = \sup {m \geq 0 : x_{r} = y_{r} in Z / p^{r} Z for every 1 \leq r \leq m},$

with $ν_{p} (x, y) = \infty$ when $x = y$ . The $p$ -adic metric on $Z_{p}$ is

$d_{p} (x, y) : = {\begin{cases} 0, & x = y, \\ p^{- ν_{p} (x, y)}, & x \neq y . \end{cases}$

Thus $d_{p} (x, y)$ is small exactly when the initial residue coordinates of $x$ and $y$ agree for a long stretch. This metric is defined directly on the compatible-tuple model of The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n.

Depends on

The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)
Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)