CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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For every prime $p$ , the multiplicative group $(Z / p Z)^{\times}$ is cyclic

Statement

For every prime $p$ , the group $(Z / p)^{\times}$ is cyclic of order $p - 1$ .

Facts & Assumptions

Given: A prime $p$ .

[L1]

$Z / p$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L2]

Every field is an integral domain (Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring).

[L3]

Every finite subgroup of the unit group of an integral domain is cyclic (Every finite subgroup of the unit group of an integral domain is cyclic).

[L4]

$(Z / p)^{\times}$ is finite of order $φ (p)$ (The unit group $(Z / n)^{\times}$ and Euler's totient $φ (n) = ∣ (Z / n)^{\times} ∣$ for $n \geq 1$ ).

[L5]

For prime $p$ , $φ (p) = p - 1$ ( $φ (1) = 1$ , and $φ (p) = p - 1$ for every prime $p$ ).

Proof

technique · direct

1.1

By [L1] and [L2], $Z / p$ is an integral domain.

L1L2

2.1

By [L4], its entire unit group is a finite subgroup of its units, so [L3] makes it cyclic; [L4] and [L5] give its order. This includes $p = 2$ , when the group is trivial.

step 1.1L3L4L5∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 91 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

William Stein, Elementary Number Theory, Theorem 2.5.8 (standard reference, not scraped)

For every prime pp, the multiplicative group (Z/pZ)×(Z/pZ)× is cyclic