An Elementary Proof That Rationally Isometric Quadratic Forms Are Isometric

Abstract

Let $R$ be a valuation ring with fraction field $K$ and $2\in R^\times$. 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 Witt's Cancelation Theorem 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. A python implementation of the algorithm derived from the proof can be found on the author's home page.