An elementary proof that rationally isometric quadratic forms are isometric

Abstract

Let R be a valuation ring with fraction field K and 2 ∈ R ×. We give an elementary proof of the following known result: two unimodular quadratic forms over R are isometric over K if and only if they are isometric over R. Our proof does not use cancelation of quadratic forms and yields an explicit algorithm to construct an isometry over R from a given isometry over K. The statement actually holds for hermitian forms over valuated involutary division rings, provided mild assumptions.