TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The canonical map from the integers into Zp is injective and has dense image

Statement

The canonical homomorphism from $Z$ to $Z_{p}$ is injective, and its image is dense in $Z_{p}$ .

Facts & Assumptions

Given: The canonical map from $Z$ to $Z_{p}$ .

[F1]

The canonical map sends $m$ to the residue tuple $(m m o d p^{n})_{n \geq 1}$ (The canonical map from Z to Zp sends an integer to its coherent residue classes modulo p^n).

[L1]

An element of $Z_{p}$ is a compatible system of residue classes modulo $p^{n}$ (The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n).

Proof

technique · direct

1.1

If an integer $m$ maps to $0$ , then [F1] says $m m o d p^{n} = 0$ for every $n \geq 1$ . Thus every power $p^{n}$ divides $m$ , which is possible only for $m = 0$ . So the canonical map is injective.

F1givenalgebra

1.2

A basic neighbourhood in $Z_{p}$ fixes some residue class modulo $p^{n}$ . Let $x = (x_{r}) \in Z_{p}$ and let $n \geq 1$ . Choose an integer $m$ representing the coordinate $x_{n} \in Z / p^{n} Z$ . Then [F1] gives that the image of $m$ has $n$ -th coordinate $x_{n}$ , and [L1] implies that this image agrees with $x$ in every earlier coordinate as well. So every basic neighbourhood of $x$ meets the embedded copy of $Z$ , which is therefore dense.

F1L1givenchoose

2.1

Step 1.1 proves injectivity and step 1.2 proves density. The zero element is treated in both arguments without any extra case split.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

4 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

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