Definition 2

Let Q be a quadratic form on E. An element x of E is said to be singular (relative to Q) if Q(x) = 0. A submodule F of E is said to be: 1) singular if there exists an element x ≠ 0 of F which is singular and orthogonal to F; 2) totally singular if the restriction of Q to F is zero.

The kernel of the quadratic module (E, Q) (§ 3, n° 4) consists of the singular elements of E^0; in order that a submodule F be singular, it is necessary and sufficient that its kernel be ≠ {0}. Since Φ(x, y) = Q(x + y) − Q(x) − Q(y), every totally singular submodule ≠ {0} is singular. Since Φ(x, x) = 2Q(x), every singular vector is isotropic and every singular submodule (resp. totally singular submodule) is isotropic (resp. totally isotropic); the converse is true if A is a field of characteristic ≠ 2. Every submodule contained in a totally singular submodule is itself totally singular. The sum of a family of pairwise orthogonal totally singular submodules is a totally singular submodule. The set of totally singular submodules of E, ordered by inclusion, is inductive; hence every totally singular submodule of E is contained in a maximal totally singular submodule.