Definition:Discriminant of Bilinear Form

Definition
Let $\mathbb K$ be a field.

Let $V$ be a vector space over $\mathbb K$ of finite dimension $n>0$.

Let $b : V\times V \to \mathbb K$ be a bilinear form on $V$.

Let $A$ be the matrix of $b$ relative to an ordered basis of $V$.

If $b$ is nondegenerate, its discriminant is the equivalence class of the determinant $\det A$ in the quotient group $\mathbb K^\times / (\mathbb K^\times)^2$.

If $b$ is degenerate, its discriminant is $0$.

Also defined as
Some authors simply do not define the discriminant of a degenerate bilinear form.

Also see

 * Discriminant of Bilinear Form is Well Defined
 * Definition:Discriminant of Quadratic Form