DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-26

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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.

Depends on

Used by

The trace form of a finite extension is nondegenerate exactly when the extension is separable Theorem

Dependency tree · two levels

7 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 2 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Section 5 (standard reference, not scraped)

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