TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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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The multiplicative group $F_{q}^{\times}$ of a finite field is cyclic

The multiplicative group $F^{\times} = F ∖ {0}$ of every finite field $F$ is cyclic.

Given: A finite field $F$ .

[L1]

A finite field has a finite underlying set (Finite fields and their order).

[L2]

[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).

technique · direct

1.1givenL1L2

By [L2], $F^{\times}$ is the unit group of the integral domain $F$ . It is finite by [L1].

2.1step 1.1L3∎

Apply [L3] to the finite subgroup $F^{\times}$ of itself to conclude that it is cyclic.

Depends on