An elementary proof that rationally isometric quadratic forms are isometric

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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.

Original languageEnglish
Pages (from-to)117-123
Number of pages7
JournalArchiv der Mathematik
Issue number2
StatePublished - Aug 2014
Externally publishedYes

Bibliographical note

Funding Information:
by a Swiss National Foundation of Science Grant no.


  • Grothendieck-Serre conjecture
  • Hermitian form
  • Quadratic form
  • Rational isomorphism
  • Valuation

ASJC Scopus subject areas

  • General Mathematics


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