DefinitionDefinition: AI-adaptedProof: Not applicablejudge 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.

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.

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

Definition

Depends on

Dependency tree · 2 levels

Sources

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

Definition

Depends on

Dependency tree · 2 levels

Sources

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