DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

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The trace form $(x, y) \mapsto {Tr}_{K / F} (x y)$ of a finite extension

Definition

Let $K / F$ be a finite field extension. The trace form of $K / F$ is the function

$T_{K / F} ⁣ : K \times K \to F, T_{K / F} (x, y) : = {Tr}_{K / F} (x y),$

where ${Tr}_{K / F}$ is the field trace (The norm $N_{K / F}$ and trace ${Tr}_{K / F}$ of a finite field extension).

Because multiplication in $K$ is commutative and the trace is $F$ -linear in its argument, the trace form is a symmetric bilinear form on the $F$ -vector space $K$ in the sense of Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms. The later theorem on this page identifies exactly when it is nondegenerate.

The trace form $(x, y) \mapsto {Tr}_{K / F} (x y)$ of a finite extension

Definition

Depends on

Dependency tree · 2 levels

Sources

The trace form (x,y)↦Tr⁡K/F(xy)(x,y)↦TrK/F​(xy) of a finite extension

Definition

Depends on

Dependency tree · 2 levels

Sources

The trace form $(x, y) \mapsto {Tr}_{K / F} (x y)$ of a finite extension