Squares / A.R. Rajwade.

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

The theorem of Hurwitz (1898) on the 2, 4, 8-identities -- The 2n-identities and the Stufe of fields : theorems of Pfister and Cassels -- Examples of the Stufe of fields and related topics -- Hilbert's 17th problem and the function fields R(X), Q(X), and R(X, Y) -- Positive semi-definite functions and sums of squares in R(X1,X2, ..., Xn) -- Introduction to Hilbert's theorem (1888) in the ring R[X1,X2, ..., Xn] -- The two proofs of Hilbert's main theorem; Hilbert's own and the other of Choi and Lam -- Theorems of Reznick and of Choi, Lam and Reznick -- Theorems of Choi, Calderon and of Robinson -- The Radon function and the theorem of Hurwitz-Radon (1922-23) -- Introduction to the teory of quadratic forms -- Theory of multiplicative forms and of Pfister forms -- The rational admissibility of the triple (r, s, n) and the Hopf condition -- Some interesting examples of bilinear identities and a theorem of Gabel -- Artin-Schreier theory of formally real fields -- Squares and sums of squares in fields and their extension fields -- Pourchet's theorem that P(Q(X)) = 5 and related results -- Examples of the Stufe and pythagroas number of fields using the Hasse-Minkowski theorem -- Reduction of matrices to canonical forms (for Chapter 10) -- Convex sets (for chaptes 6,7,8,9).

Many classical and modern results and quadratic forms are brought together in this book. The treatment is self-contained and of a totally elementary nature requiring only a basic knowledge of rings, fields, polynomials, and matrices, such that the works of Pfister, Hilbert, Hurwitz and others are easily accessible to non-experts and undergraduates alike. The author deals with many different approaches to the study of squares; from the classical works of the late 19th century, to areas of current research. Anyone with an interest in algebra or number theory will find this a most fascinating volume.