DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-04

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The p-adic numbers as a metric completion

Definition

Let $p$ be a prime. The space of $p$ -adic numbers $Q_{p}$ is the Cauchy-sequence completion constructed in Every metric space has a completion, constructed as the equivalence classes of its Cauchy sequences for the metric

$d_{p} (x, y) : = ∣ x - y ∣_{p},$

where $∣ \cdot ∣_{p}$ is the absolute value of The p-adic absolute value on the rationals. Thus its points are equivalence classes of rational $d_{p}$ -Cauchy sequences, two sequences being equivalent when their termwise distance tends to $0$ . The completion datum is taken in the sense of A completion of a metric space: a complete metric space together with an isometric embedding onto a dense subspace, so $Q$ comes equipped with its named dense isometric embedding into $Q_{p}$ . The next theorem equips this complete metric space with the field operations extending those of $Q$ .

Depends on

Used by

Dependency tree · two levels

26 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

Andrew V. Sutherland, 18.782 Lecture 8 (standard reference, not scraped)
J. S. Milne, Algebraic Number Theory, Chapter 7 (standard reference, not scraped)