Squares [Paperback]

Many classical and modern results and quadratic forms are brought together in this book. The author deals with many different approaches to the study of squares.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 and only a basic knowledge of rings, fields, polynomials, and matrices is required.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 and only a basic knowledge of rings, fields, polynomials, and matrices is required.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.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 HurwitzRadon; 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. ArtinSchreier 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 HasseMinkowski l“„