The Algebras of Quadratic Forms

  • O. Timothy O’Meara
Part of the Classics in Mathematics book series (volume 117)


Our purpose in this chapter is to introduce three algebras of importance in the theory of quadratic forms, the Clifford algebra, the quaternion algebra, and the Hasse algebra. The Clifford algebra will be developed from first principles and its main use for us will be in the definition of an invariant called the spinor norm. The quaternion algebra and the Hasse algebra play an important role in the arithmetic theory of quadratic forms. The definition of the Hasse algebra depends on some of the structure theory of central simple algebras, in particular it needs Wedderburn’s theorem and the theory of similarity of algebras that is normally used in defining the Brauer group. We have therefore included a proof of Wedderburn’s theorem and some of its consequences. Also included as a convenience to the reader is a brief discussion of the tensor product of finite dimensional vector spaces1.


Quadratic Form Tensor Product Left Ideal Division Algebra Clifford Algebra 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • O. Timothy O’Meara
    • 1
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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