Quaternion Algebras I

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


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.


Number Field Division Algebra Invertible Element Quadratic Extension Quaternion Algebra 
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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.zbMATHCrossRefGoogle 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.zbMATHGoogle Scholar
  6. Reiner, I. (1975). Maximal Orders. Academic Press, London.zbMATHGoogle Scholar
  7. Platonov, V. and Rapinchuk, A. (1994). Algebraic Groups and Number Theory. Academic Press, London.zbMATHGoogle Scholar
  8. Elstrodt, J., Grunewald, F., and Mennicke, J. (1987). Vahlen’s group of Clifford matrices and spin-groups. Math. Zeitschrift, 196: 369–390.MathSciNetzbMATHCrossRefGoogle 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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