False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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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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FALSE: the trace map of every finite extension is surjective

Statement

False claim: the trace map of every finite field extension is surjective.

Facts & Assumptions

Given: The extension $F_{p} (t) / F_{p} (t^{p})$ .

[L1]

For this purely inseparable extension, the trace map is identically zero (For $F_{p} (t) / F_{p} (t^{p})$ , the trace is identically zero).

Refutation

technique · direct

1.1

By [L1], the trace map ${Tr}_{F_{p} (t) / F_{p} (t^{p})} ⁣ : F_{p} (t) \to F_{p} (t^{p})$ has image ${0}$ .

2.1

The codomain is not the zero field, so this map is not surjective. Therefore the displayed claim is false.

step 1.1algebra∎

Depends on

For $\mathbb F_p(t)/\mathbb F_p(t^p)$, the trace is identically zero

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

B. Conrad, Norm and trace, Theorem 2.5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Remark 5.47 (standard reference, not scraped)