ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The subfields of $F_{64}$ have orders $2, 4, 8, 64$

Example

The subfields of $F_{64} = F_{2^{6}}$ have orders $2, 4, 8, 64$ . For $d = 1, 2, 3, 6$ , the unique subfield of order $2^{d}$ is

${a \in F_{64} : a^{2^{d}} = a} .$

Facts & Assumptions

Given: A field $F$ of order $64 = 2^{6}$ .

[L1]

The subfields of $F_{p^{n}}$ are the unique fields of order $p^{d}$ for positive divisors $d$ of $n$ , and each is the root set of $t^{p^{d}} - t$ (The subfields of $F_{p^{n}}$ are the unique fields $F_{p^{d}}$ for positive divisors $d$ of $n$ ).

Verification

technique · direct

1.1givenalgebra

The positive divisors of $6$ are $1, 2, 3, 6$ .

2.1step 1.1L1algebra∎

Apply [L1] with $p = 2$ and these four divisors to obtain the orders $2^{1}, 2^{2}, 2^{3}, 2^{6}$ , namely $2, 4, 8, 64$ , and the displayed root-set descriptions.

Depends on

The subfields of $\mathbb F_{p^n}$ are the unique fields $\mathbb F_{p^d}$ for positive divisors $d$ of $n$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

K. Conrad, Finite Fields, Theorem 2.8 (standard reference, not scraped)

The subfields of F64F64​ have orders 2,4,8,642,4,8,64