Quadratic Forms

  • Herbert Gross
Part of the Progress in Mathematics book series (PM, volume 1)


Quadratic forms are closely related to orthosymmetric sesquilinear forms and, to a large extent, they behave very similarly. In fact, the two concepts partly overlap (cf. Example 2 in Section 3 below). For the purpose of illustration we start with the classical notion of a quadratic form
$$Q:E \to K$$
on a k-vector space E over a commutative field k of arbitrary characteristic. The map Q is called a quadratic form if 1) we have Q(λx) = λ2Q(x) for all λ ∈ k, x ∈ E, and 2) the assignment Ψ: (x, y) ⟼ Q(x+y) - Q(x) - Q(y) from E × E into k is bilinear (Ψ is called the bilinear form associated to Q; it is, by necessity, a symmetric form). Thus, by definition, we have the formula Q(x+y) = Q(x) + Q(y) + Ψ (x, y).


Quadratic Form Singular Vector Division Ring Hyperbolic Plane Hermitean Form 
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References to Chapter XIV

  1. [1]
    Artin AAAA, Geometric algebra. Interscience Publ., New York 1957.Google Scholar
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    N. Bourbaki, Formes sesquilinéaires et formes quadratiques. ASI 1272, Hermann Paris, 1959.Google Scholar
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We have not touched upon the definition of the Arf invariant of finite dimensional quadratic spaces. Besides [3], which brings a definition of the invariant taylored to the concept of quadratic form as discussed here, we add below a list of further references bearing on the topic (cf. Appendix I to Chapter XVI).

  1. [5]
    J. Dieudonné, Pseudo-discriminant and Dickson invariant. Pacific J. Math. 5 (1955) 907–910.CrossRefGoogle Scholar
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    R. H. Dye, On the Arf Invariant. J. Algebra 53 (1978) 36–39.CrossRefGoogle Scholar
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    N. Jacobson, Clifford algebras for algebras with involution of type D. J. Algebra 1 (1964) 288–300.CrossRefGoogle Scholar
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    I. Kaplansky, Linear Algebra and Geometry. Allyn & Bacon, Boston 1969, page 29.Google Scholar
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    I. Kaplansky and R.J. Shaker, Abstract Quadratic Forms. Canad. J. Math. 21 (1969) 1218–1233.Google Scholar
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    W. Klingenberg and E. Witt, Ueber die Arfsehe Invariante quadratischer Formen mod 2. J. reine angew. Math. 193 (1954) 121–122.Google Scholar
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    M. Kneser, Bestimmung des Zentrums der Cliffordschen Algebra. J. reine angew. Math. 193 (1954) 123–124.Google Scholar
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    T. A. Springer, Note on quadratic forms in characteristic 2. Nieuw archief voor wiskunde (3), X (1962) 1–10.Google Scholar
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References to Appendix I

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    A. A. Albert, Structure of Algebras. (3rd print. of rev. ed.) AMS Coll. Publ. XXIV, NY 1968.Google Scholar
  2. [2]
    M. Deuring, Algebren. (Zweite, korrigierte Auflage), Ergebnisse der Math. vol 41, Springer Berlin 1968.CrossRefGoogle Scholar
  3. [3]
    I. N. Herstein, Noncommutative Rings. Carus Math. Monographs 15, Math. Assoc. of America 1968, distr. by J. Wiley & Sons, Inc.Google Scholar

Copyright information

© Springer Science+Business Media New York 1979

Authors and Affiliations

  • Herbert Gross
    • 1
  1. 1.Mathematisches InstitutUniversität ZürichZürichSwitzerland

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