Pub. Date:
Cambridge University Press


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This work is a self-contained treatise on the research conducted on squares by Pfister, Hilbert, Hurwitz, and others. Many classical and modern results and quadratic forms are brought together in this book, and the treatment requires only a basic knowledge of rings, fields, polynomials, and matrices. The author deals with many different approaches to the study of squares, from the classical works of the late nineteenth century, to areas of current research.

Product Details

ISBN-13: 9780521426688
Publisher: Cambridge University Press
Publication date: 11/28/2004
Series: London Mathematical Society Lecture Note Series , #171
Pages: 286
Product dimensions: 5.98(w) x 8.98(h) x 0.67(d)

Table of Contents

1. The theorem of Hurwitz; 2. The 2n theorems and the Stufe of fields; 3. Examples of the Stufe of fields and related topics; 4. Hilbert's 17th problem; 5. Positive definite functions and sums of squares; 6. An introduction to Hilbert's theorem; 7. The two proofs of Hilbert's theorem; 8. Theorems of Reznick and Choi, Lam and Reznick; 9. Theorems of Choi, Calderon and Robinson; 10. The theorem of Hurwitz-Radon; 11. An introduction to quadratic form theory; 12. The theory of multiplicative forms and Pfister forms; 13. The Hopf condition; 14. Examples of bilinear identities and a theorem of Gabel; 15. Artin-Schreier theory of formally real fields; 16. Squares and sums of squares in fields and their extension fields; 17. Pourchet's theorem and related results; 18. Examples of the Stufe and Pythagoras number of fields using the Hasse-Minkowski theorem; Appendix: Reduction of matrices to canonical form.

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