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Quadratic Forms

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

Abstract

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).

Keywords

Quadratic Form Singular Vector Division Ring Hyperbolic Plane Hermitean Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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).

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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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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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