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 $\dfrac {\mathbb K^\times} {\paren {\mathbb K^\times}^2}$.

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

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.