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.
- Grothendieck-Serre conjecture
- Hermitian form
- Quadratic form
- Rational isomorphism
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