Quaternion Algebras I

  • Colin Maclachlan
  • Alan W. Reid
Part of the Graduate Texts in Mathematics book series (GTM, volume 219)

Abstract

Throughout this book, the main algebraic structure which plays a major role in all investigations is that of a quaternion algebra over a number field. In this chapter, the basic theory of quaternion algebras over a field of characteristic ≠ 2 will be developed. This will suffice for applications in the following three chapters, but a more detailed analysis of quaternion algebras will need to be developed in order to appreciate the number-theoretic input in the cases of arithmetic Kleinian groups. This will be carried out in a later chapter. For the moment, fundamental elementary notions for quaternion algebras are developed. In this development, use is made of two key results on central simple algebras and these are proved independently in the later sections of this chapter.

Keywords

Number Field Division Algebra Invertible Element Quadratic Extension Quaternion Algebra 
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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Further Reading

  1. Vignéras, M.-F. (1980a). Arithmétique des Algèbres de Quaternions..Lecture Notes in Mathematics No. 800. Springer-Verlag, Berlin.Google Scholar
  2. Lam, T. (1973). The Algebraic Theory of Quadratic Forms. W.A. Benjamin, Reading, MA.Google Scholar
  3. O’Meara, O. (1963). Introduction to Quadratic Forms. Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  4. Pierce, R. (1982). Associative Algebra. Graduate Texts in Mathematics Vol. 88. Springer-Verlag, New York.CrossRefGoogle Scholar
  5. Cohn, P. (1991). Algebra, Volume 3. Wiley, Chichester.MATHGoogle Scholar
  6. Reiner, I. (1975). Maximal Orders. Academic Press, London.MATHGoogle Scholar
  7. Platonov, V. and Rapinchuk, A. (1994). Algebraic Groups and Number Theory. Academic Press, London.MATHGoogle Scholar
  8. Elstrodt, J., Grunewald, F., and Mennicke, J. (1987). Vahlen’s group of Clifford matrices and spin-groups. Math. Zeitschrift, 196: 369–390.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Colin Maclachlan
    • 1
  • Alan W. Reid
    • 2
  1. 1.Department of Mathematical Sciences, Kings CollegeUniversity of AberdeenAberdeenUK
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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