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Unformatted text preview: Quaternion Algebras and Quadratic Forms by Zi Yang Sham A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Pure Mathematics Waterloo, Ontario, Canada, 2008 c Zi Yang Sham 2008 Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be electronically available to the public. ii Abstract The main goal of this Masters thesis is to explore isomorphism types of quater- nion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theo- rem proved by Merkurjev in 1981. iii Acknowledgements I need to thank my supervisor Professor David McKinnon for his kind guidance throughout the duration of my Masters degree, and Professor Rahim Moosa and Professor Ken Davidson for being my readers. I would like to thank every instructor in the University of Waterloo who taught me for the past five years, without all of you I could not possibly be finishing this degree in pure mathe- matics. I am also grateful to Lalit Jain and Collin Roberts for helping me with typesetting. Finally my thanks go to my parents and Yunzhi for their loving support and concern. iv Dedication To everyone who likes pure mathematics v Contents 1 Quadratic Forms 1 1.1 Quadratic Forms and Quadratic Spaces . . . . . . . . . . . 1 1.2 Diagonalization of Quadratic Forms . . . . . . . . . . . . . . 7 1.3 Hyperbolic Plane and Hyperbolic Spaces . . . . . . . . . . 13 1.4 Witts Decomposition and Cancellation . . . . . . . . . . . 16 2 Quaternion Algebras 22 2.1 Basic Properties of Quaternion Algebras . . . . . . . . . . . . . . 22 2.2 Determining the Isomorphism Type . . . . . . . . . . . . . . . . 24 2.3 Quaternion Algebras over Different Fields . . . . . . . . . . . . . 33 3 The Brauer Group and the Theorem of Merkurjev 37 3.1 Properties of the Brauer Group . . . . . . . . . . . . . . . . . . . 37 3.2 The Role of Quaternion Algebras in the Brauer Group . . . . . . 40 3.3 The Theorem of Merkurjev . . . . . . . . . . . . . . . . . . . . . 42 4 Characterization of Quaternion Algebras 44 4.1 Three Similar Theorems . . . . . . . . . . . . . . . . . . . . . 44 4.2 Properties of Real Closed Fields . . . . . . . . . . . . . . . . 47 4.3 R need not be in the center of D . . . . . . . . . . . . . . . . 52 4.4 A Few Lemmas and the Proof . . . . . . . . . . . . . . . . . 53 References 61 vi Chapter 1 Quadratic Forms (From Chapter I of [8]) 1.1 Quadratic Forms and Quadratic Spaces An n-ary quadratic form (i.e. a 2-form) over a field F is a polynomial f in n variables over F that is homogeneous of degree 2. (Please note that throughout this article, the characteristic of F is assumed not to be 2.) It has the generalis assumed not to be 2....
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second2 - Quaternion Algebras and Quadratic Forms by Zi...

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