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

1.1F1givenalgebra

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.

1.2F1L1givenchoose

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.

2.1step 1.1step 1.2∎

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

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)