Algebraic theory of quadratic numbers /

In the Scylla and Charybdis of mathematical exposition, too much emphasis on examples and calculations can obscure the synthesis and too much emphasis on theory may leave sophisticated students still unable to solve simple problems if answered only implicitly by the theory. So creating meaning involves building bridges between the short view and the long view. Here, Trifkovic (Univ. of Victoria, Canada) introduces algebraic number theory with all the usual abstractions, except limited to whatever proves necessary just for extensions of degree two. Degree two, in particular, makes the role of Galois theory "invisible," as the author terms it. Though many books offer study of quadratic forms and Pell's equation by purely elementary means, the approach here strikes a perfect balance, achieving legible results while preparing students for deeper study. A "crash course" in ring theory obviates abstract algebra prerequisites, and the author gives a similarly generous review and amplification of linear algebra tools. Degree two allows for many illuminating diagrams, an algebraic number theory rarity. In 2004, M. Bhargava made headlines by generalizing Gauss's classical composition laws for quadratic forms; remarkably, in the culminating chapter here, some of that work receives undergraduate-targeted exposition. Summing Up: Recommended. Upper-division undergraduates and above. D. V. Feldman University of New Hampshire