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An elementary proof that rationally isometric quadratic forms are isometric

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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.

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Authors and Affiliations

  1. Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

    Uriya A. First

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  1. Uriya A. First
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Correspondence to Uriya A. First.

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This research was supported by a Swiss National Foundation of Science Grant no. IZK0Z2_151061.

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First, U.A. An elementary proof that rationally isometric quadratic forms are isometric. Arch. Math. 103, 117–123 (2014). https://doi.org/10.1007/s00013-014-0676-7

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  • DOI: https://doi.org/10.1007/s00013-014-0676-7

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