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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.

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This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication by i. It is natural to consider other kinds of complex multiplications and here we shall make a start on the case when the ring of complex multiplications is isomorphic to the ring Z[σ], where σ2 = − 2.

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