DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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The norm $N_{K / F}$ and trace ${Tr}_{K / F}$ of a finite field extension

Definition

Let $K / F$ be a finite field extension, so $K$ is a finite-dimensional $F$ -vector space of degree $[K : F]$ (The degree $[K : F] = \dim_{F} K$ of a finite field extension). For $a \in K$ , let

$m_{a} ⁣ : K \to K, x \mapsto a x,$

the $F$ -linear operator of multiplication by $a$ .

The norm and trace of $a$ from $K$ to $F$ are

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

where determinant and trace are those of the published linear-operator notions (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).

Because $K$ is a field, $m_{a}$ is the zero operator exactly when $a = 0$ and is an automorphism exactly when $a \neq 0$ . Later items identify these two quantities with the classical embedding formulas and the trace form.

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

J. S. Milne, Fields and Galois Theory, v5.10, Chapter 6, Section 5 (standard reference, not scraped)
B. Conrad, Norm and trace, Sections 1-3 (standard reference, not scraped)

The norm NK/FNK/F​ and trace Tr⁡K/FTrK/F​ of a finite field extension