DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

A quadratic form $q$ in arbitrary characteristic and its polar form $b_{q} (u, v) = q (u + v) - q (u) - q (v)$

Definition

Let $V$ be a vector space over a field $F$ . A quadratic form on $V$ is a function $q : V \to F$ such that

$q (a v) = a^{2} q (v)$

for all $a \in F$ and $v \in V$ , and such that its polar form

$b_{q} (u, v) : = q (u + v) - q (u) - q (v)$

is bilinear. This definition is valid in every characteristic. In characteristic $2$ , the polar form need not determine $q$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Bilinear Forms, §7 (standard reference, not scraped)
H. Pinkham, Linear Algebra, Chapter 7 (standard reference, not scraped)

A quadratic form qq in arbitrary characteristic and its polar form bq(u,v)=q(u+v)−q(u)−q(v)bq​(u,v)=q(u+v)−q(u)−q(v)