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In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form ''Q'' over a field ''K'' takes values in the Brauer group Br(''K''). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt. The quadratic form ''Q'' may be taken as a diagonal form :Σ ''a''''i''''x''''i''2. Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras :(''a''''i'', ''a''''j'') for ''i'' < ''j''. This is independent of the diagonal form chosen to compute it.〔Lam (2005) p.118〕 It may also be viewed as the second Stiefel–Whitney class of ''Q''. ==Symbols== The invariant may be computed for a specific symbol φ taking values ±1 in the group ''C''2.〔Milnor & Husemoller (1973) p.79〕 In the context of quadratic forms over a local field, the Hasse invariant may be defined using the Hilbert symbol, the unique symbol taking values in ''C''2.〔Serre (1973) p.36〕 The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.〔Serre (1973) p.39〕 For quadratic forms over a number field, there is a Hasse invariant ±1 for every finite place. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.〔Conner & Perlis (1984) p.16〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hasse invariant of a quadratic form」の詳細全文を読む スポンサード リンク
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