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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FALSE: for every finite extension, the norm is just the product over the embeddings

Statement

False claim: for every finite field extension $K / F$ and every $a \in K$ , the norm $N_{K / F} (a)$ is just the product of the distinct $F$ -embeddings of $a$ .

Facts & Assumptions

Given: The purely inseparable extension $F_{p} (t) / F_{p} (t^{p})$ and the element $t$ .

[L1]

In a finite extension, the norm is the product over embeddings raised to the inseparable degree (Norm and trace from embeddings, with the inseparable exponent in the norm formula).

[L2]

The extension $F_{p} (t) / F_{p} (t^{p})$ is purely inseparable of degree $p$ and has only the inclusion embedding (For $F_{p} (t) / F_{p} (t^{p})$ , the trace is identically zero).

Refutation

technique · direct

1.1

By [L2], the product over the distinct embeddings of $t$ is just $t$ itself.

2.1

But [L1] and [L2] give $N_{F_{p} (t) / F_{p} (t^{p})} (t) = t^{p},$ and $t^{p} \neq t$ in the rational function field. Therefore the displayed claim is false.

step 1.1L1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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, Theorems 2.3 and 3.2 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Remark 5.47 (standard reference, not scraped)