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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Further Reading
Vignéras, M.-F. (1980a). Arithmétique des Algèbres de Quaternions..Lecture Notes in Mathematics No. 800. Springer-Verlag, Berlin.
Lam, T. (1973). The Algebraic Theory of Quadratic Forms. W.A. Benjamin, Reading, MA.
O’Meara, O. (1963). Introduction to Quadratic Forms. Springer-Verlag, Berlin.
Pierce, R. (1982). Associative Algebra. Graduate Texts in Mathematics Vol. 88. Springer-Verlag, New York.
Cohn, P. (1991). Algebra, Volume 3. Wiley, Chichester.
Reiner, I. (1975). Maximal Orders. Academic Press, London.
Platonov, V. and Rapinchuk, A. (1994). Algebraic Groups and Number Theory. Academic Press, London.
Elstrodt, J., Grunewald, F., and Mennicke, J. (1987). Vahlen’s group of Clifford matrices and spin-groups. Math. Zeitschrift, 196: 369–390.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Maclachlan, C., Reid, A.W. (2003). Quaternion Algebras I. In: The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics, vol 219. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-6720-9_3
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6720-9_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3122-1
Online ISBN: 978-1-4757-6720-9
eBook Packages: Springer Book Archive