Quadratic form - Knowlpedia

Definition

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

q(av)=a^2q(v)

for all $a\in k$ , $v\in V$ , and the polarization

B_q(v,w)=q(v+w)-q(v)-q(w)

is a bilinear form. The pair $(V,q)$ is called a quadratic space.

Characteristic not two

If $2$ is invertible in $k$ , then $B_q$ is symmetric and

q(v)=\frac12B_q(v,v).

Thus quadratic forms and symmetric bilinear forms determine one another in this setting. In characteristic $2$ , the quadratic form contains information not recoverable from its polarization.

Polar radical and nondegeneracy

The polar radical of $q$ is the radical of $B_q$ :

\operatorname{rad}(B_q) =\{v\in V:B_q(v,w)=0\text{ for every }w\in V\}.

When $2$ is invertible, $q$ is called nondegenerate exactly when this radical is zero. In characteristic $2$ , authors use several related notions—regular, nonsingular, and nondegenerate—and may also require conditions on the restriction of $q$ to its polar radical. Statements in that setting must specify the convention rather than referring to “the radical of $q$ ” without qualification.

References

T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society, 2005. DOI record. Relevant: Chapter I.