ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

In $F_{q^{n}} / F_{q}$ , norm and trace are the Frobenius product and sum

Example

Let $F_{q^{n}} / F_{q}$ be a finite-field extension, and let $α \in F_{q^{n}}$ . Then

$N_{F_{q^{n}} / F_{q}} (α) = α α^{q} \dots α^{q^{n - 1}} = α^{1 + q + \dots + q^{n - 1}},$

and

${Tr}_{F_{q^{n}} / F_{q}} (α) = α + α^{q} + \dots + α^{q^{n - 1}} .$

Facts & Assumptions

Given: A finite extension $F_{q^{n}} / F_{q}$ and an element $α \in F_{q^{n}}$ .

[L1]

The extension is finite Galois with cyclic Galois group generated by the Frobenius $σ_{q} (x) = x^{q}$ (A finite extension of a finite field of order $q$ is Galois with cyclic Galois group generated by $x \mapsto x^{q}$ ).

[L2]

Norm and trace are the product and sum of the distinct embeddings in the separable case (Norm and trace from embeddings, with the inseparable exponent in the norm formula).

Verification

technique · direct

1.1

By [L1], the $n$ distinct $F_{q}$ -embeddings are $1, σ_{q}, \dots, σ_{q}^{n - 1}$ .

2.1

Applying [L2] to the embeddings of step 1.1 gives $N_{F_{q^{n}} / F_{q}} (α) = \prod_{i = 0}^{n - 1} σ_{q}^{i} (α) = \prod_{i = 0}^{n - 1} α^{q^{i}} = α^{1 + q + \dots + q^{n - 1}},$ and ${Tr}_{F_{q^{n}} / F_{q}} (α) = \sum_{i = 0}^{n - 1} σ_{q}^{i} (α) = α + α^{q} + \dots + α^{q^{n - 1}} .$

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

B. Conrad, Norm and trace, Section 1 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, finite fields and norm/trace (standard reference, not scraped)

In Fqn/FqFqn​/Fq​, norm and trace are the Frobenius product and sum