TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Field norm and trace agree with the determinant and trace of multiplication by an element

Statement

Let $K / F$ be a finite extension and let $a \in K$ . If

$m_{a} ⁣ : K \to K, x \mapsto a x,$

is the $F$ -linear multiplication operator, then

$N_{K / F} (a) = \det (m_{a}), {Tr}_{K / F} (a) = tr (m_{a}),$

where the right-hand side uses the published linear-operator determinant and trace.

Facts & Assumptions

Given: A finite field extension $K / F$ , an element $a \in K$ , and the operator $m_{a}$ of multiplication by $a$ .

[F1]

The field norm and trace were defined by $N_{K / F} (a) : = \det (m_{a})$ and ${Tr}_{K / F} (a) : = tr (m_{a})$ (The norm $N_{K / F}$ and trace ${Tr}_{K / F}$ of a finite field extension).

[F2]

The determinant and trace of an endomorphism are the basis-independent linear-algebra notions of the earlier pages (The determinant of an endomorphism of a finite-dimensional vector space: its matrix determinant in an ordered basis in positive dimension, and $1$ on the zero space, The basis-independent trace of an endomorphism of a finite-dimensional vector space).

Proof

technique · direct

1.1F1F2

The two displayed identities are exactly the definitions of [F1], and [F2] identifies the determinant and trace on the right with the published operator notions. Since a field extension has positive degree, the zero-dimensional determinant convention never needs a separate case here.

2.1step 1.1∎

This proves the stated dictionary identification.

Remarks

This is the promised dictionary item. The page is not introducing a second unrelated determinant or trace: the field-theoretic norm and trace are built from the same linear-algebra invariants already established for endomorphisms.

Depends on

Used by

Dependency tree · two levels

9 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, Section 5 (standard reference, not scraped)