**History of Mathematics**

Volume: 32;
2007;
336 pp;
Softcover

MSC: Primary 01;

**Print ISBN: 978-0-8218-6904-8
Product Code: HMATH/32.S**

List Price: $74.00

AMS Member Price: $59.20

MAA Member Price: $66.60

**Electronic ISBN: 978-1-4704-1808-3
Product Code: HMATH/32.E**

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

#### Supplemental Materials

# Episodes in the History of Modern Algebra (1800–1950)

Share this page *Edited by *
*Jeremy J. Gray; Karen Hunger Parshall*

A co-publication of the AMS and the London Mathematical Society

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.

#### Readership

Graduate students and research mathematicians interested in the history of mathematics and algebra.

#### Reviews & Endorsements

This book offers new light on the development and history of modern algebra.

-- EMS Newsletter

#### Table of Contents

# Table of Contents

## Episodes in the History of Modern Algebra (1800-1950)

- Cover Cover11
- Half Title i2
- Title Page iii4
- Contents v6
- Acknowledgments 110
- Introduction 312
- Babbage and French Idéologie: Functional equations, language, and the analytical method 1322
- “Very full of symbols”: Duncan F. Gregory, the calculus of operations, and the Cambridge Mathematical Journal 4958
- Divisibility theories in the early history of commutative algebra and the foundations of algebraic geometry 7382
- Kronecker’s fundamental theorem of general arithmetic 107116
- 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 117126
- Minkowski, Hensel, and Hasse: On the beginnings of the local-global principle 153162
- Research in algebra at the University of Chicago: Leonard Eugene Dickson and A. Adrian Albert 179188
- Emmy Noether’s 1932 ICM lecture on noncommutative methods in algebraic number theory 199208
- From Algebra (1895) to Moderne Algebra (1930): Changing conceptions of a discipline—A guided tour using the Jahrbuch über die Fortschritte der Mathematik 221230
- A historical sketch of B. L. van der Waerden’s work in algebraic geometry: 1926–1946 245254
- On the arithmetization of algebraic geometry 285294
- The rising sea: Grothendieck on simplicity and generality 301310
- Back Cover Back Cover1346