Episodes in the History of Modern Algebra (1800–1950)
About this Title
Jeremy J. Gray, The Open University, Milton Keynes, England and Karen Hunger Parshall, University of Virginia, Charlottesville, VA, Editors
Publication: History of Mathematics
Publication Year: 2007; Volume 32
ISBNs: 978-0-8218-6904-8 (print); 978-1-4704-1808-3 (online)
MathSciNet review: MR2307989
MSC: Primary 00B25; Secondary 01A05, 01A55, 01A60, 01A70, 13-03
Algebra, as a subdiscipline of mathematics, arguably has a history going back some 4000 years to ancient Mesopotamia. The history, however, of what is recognized today as high school algebra is much shorter, extending back to the sixteenth century, while the history of what practicing mathematicians call "modern algebra" is even shorter still.
The present volume provides a glimpse into the complicated and often convoluted history of this latter conception of algebra by juxtaposing twelve episodes in the evolution of modern algebra from the early nineteenth-century work of Charles Babbage on functional equations to Alexandre Grothendieck's mid-twentieth-century metaphor of a “rising sea” in his categorical approach to algebraic geometry. In addition to considering the technical development of various aspects of algebraic thought, the historians of modern algebra whose work is united in this volume explore such themes as the changing aims and organization of the subject as well as the often complex lines of mathematical communication within and across national boundaries. Among the specific algebraic ideas considered are the concept of divisibility and the introduction of non-commutative algebras into the study of number theory and the emergence of algebraic geometry in the twentieth century.
The resulting volume is essential reading for anyone interested in the history of modern mathematics in general and modern algebra in particular. It will be of particular interest to mathematicians and historians of mathematics.
Graduate students and research mathematicians interested in the history of mathematics and algebra.
Table of Contents
- Babbage and French Idéologie: Functional equations, language, and the analytical method
- “Very full of symbols”: Duncan F. Gregory, the calculus of operations, and the Cambridge Mathematical Journal
- Divisibility theories in the early history of commutative algebra and the foundations of algebraic geometry
- Kronecker’s fundamental theorem of general arithmetic
- Developments in the theory of algebras over number fields: A new foundation for the Hasse norm residue symbol and new approaches to both the Artin reciprocity law and class field theory
- Minkowski, Hensel, and Hasse: On the beginnings of the local-global principle
- Research in algebra at the University of Chicago: Leonard Eugene Dickson and A. Adrian Albert
- Emmy Noether’s 1932 ICM lecture on noncommutative methods in algebraic number theory
- From Algebra (1895) to Moderne Algebra (1930): Changing conceptions of a discipline—A guided tour using the Jahrbuch über die Fortschritte der Mathematik
- A historical sketch of B. L. van der Waerden’s work in algebraic geometry: 1926–1946
- On the arithmetization of algebraic geometry
- The rising sea: Grothendieck on simplicity and generality