This book offers new light on the development and history of modern algebra. The book brings together suitably revised, chapter-length versions of twelve lectures that were given at the workshop on the history of 19th and 20th century algebra held at the Mathematical Sciences Research Institute in Berkeley in 2003.

The introduction and chapters written by prominent mathematicians and historians of mathematics (J. J. Gray, K. H. Parshall, E. L. Ortiz, S. E. Despeaux, O. Neumann, H. M. Edwards, G. Frei, J. Schwermer, D. D. Fenster, Ch. W. Curtis, L. Corry, N. Schappacher, S. Slembek, C. McLarty) provide complex and detailed overviews of the evolution of modern algebra from the early 19th century work of Ch. Babbage on function equations through to the description of the development of calculus operations (British research done before and documented within the Cambridge Mathematical Journal), the analysis of divisibility theories in the early history of commutative algebra (contributions by C. F. Gauss, E. E. Kummer, E. I. Zolotarev, L. Kronecker, D. Hilbert, E. Noether, etc.), a presentation of Kronecker's fundamental theorem on general arithmetic, a description of advances in the theory of algebras and algebraic number theory (works by J. H. M. Wedderburn, A. Hurwitz, R. Brauer, H. Hasse, E. Noether, H. Minkowski, K. Hensel, L. Dickson, A. A. Albert, etc.) and in the analysis of the foundations of algebraic geometry and its arithmetization and development (results of H. Hasse, E. Noether, F. Severi, A. Grothendieck, etc.).

The topics discussed represent the long and difficult process of the changing organization of the main subjects, the changing algebraic themes, ideas and concepts, as well as the results of mathematical communication and collaboration within and across national boundaries. A comprehensive list of references as well as many detailed notes are added at the end of each chapter. The book can be recommended to readers who are interested in the history of modern mathematics in general and modern algebra in particular. It is suitable for mathematicians, historians of mathematics and science, teachers and students.

Reviewer:

mbec