Numbers and symmetry: an introduction to algebra (Q2785504)

This is a well written and engaging text, dealing mainly with the elements of algebraic number theory, suitable for second year university students in mathematics. There are plenty of exercises, and useful end of chapter notes emphasizing historical aspects of the topics concerned. NEWLINENEWLINENEWLINENumber and symmetry lie at the root of modern algebra, and the authors emphasize the development of algebraic concepts through successive extensions of number systems. The first five chapters deal with fundamental concepts, such as the Euclidean algorithm, units and primes, applied to the commutative rings \(\mathbb{Z}\), \(\mathbb{Z}_n\), \(\mathbb{Z}[i]\), \(\mathbb{Z}[\sqrt2]\), and polynomial rings in one variable. Symmetry is the study of structure-preserving transformations, and so the next four chapters deal with groups of transformations, leading to the analysis of wallpaper patterns, where there is a thorough discussion of one and two dimensional lattices, including frieze patterns. The last three chapters are on finite fields, linear algebra and error-correcting codes, respectively. There is also a good description of the design of compact discs to illustrate these concepts. NEWLINENEWLINENEWLINEAlthough the subject matter can be very abstract, the exposition is kept at such a concrete level that the beginner should not feel lost. In a mathematical textbook, the truth of a theorem should be clearer after the proof than it was before, and the authors make a very good effort to this end.