# Witt theorem

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Any isometry between two subspaces and of a finite-dimensional vector space , defined over a field of characteristic different from 2 and provided with a metric structure induced from a non-degenerate symmetric or skew-symmetric bilinear form , may be extended to a metric automorphism of the entire space . The theorem was first obtained by E. Witt .
Witt's theorem may also be proved under wider assumptions on and , . In fact, the theorem remains valid if is a skew-field, is a finite-dimensional left -module and is a non-degenerate -Hermitian form (with respect to some fixed involutory anti-automorphism of , cf. Hermitian form) satisfying the following condition: For any there exists an element such that (property ). Property holds if, for example, is a Hermitian form and the characteristic of is different from 2, or if is an alternating form. Witt's theorem is also valid if is a field and is the symmetric bilinear form associated with a non-degenerate quadratic form on . It follows from Witt's theorem that the group of metric automorphisms of transitively permutes the totally-isotropic subspaces of the same dimension and that all maximal totally-isotropic subspaces in have the same dimension (the Witt index of ). A second consequence of Witt's theorem may be stated as follows: The isometry classes of non-degenerate symmetric bilinear forms of finite rank over with direct orthogonal sum form a monoid with cancellation; the canonical mapping of this monoid into its Grothendieck group is injective. The group is called the Witt–Grothendieck group of ; the tensor product of forms induces on it the structure of a ring, which is known as the Witt–Grothendieck of .