A History of Abstract Algebra

Front Cover
Springer Science & Business Media, Sep 20, 2007 - Mathematics - 168 pages

Prior to the nineteenth century, algebra meant the study of the solution of polynomial equations. By the twentieth century algebra came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields. This presentation provides an account of the history of the basic concepts, results, and theories of abstract algebra.

The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing context from which the reader may gain a deeper appreciation of the mathematics involved.

Key features:

* Begins with an overview of classical algebra

* Contains separate chapters on aspects of the development of groups, rings, and fields

* Examines the evolution of linear algebra as it relates to other elements of abstract algebra

* Highlights the lives and works of six notables: Cayley, Dedekind, Galois, Gauss, Hamilton, and especially the pioneering work of Emmy Noether

* Offers suggestions to instructors on ways of integrating the history of abstract algebra into their teaching

* Each chapter concludes with extensive references to the relevant literature

Mathematics instructors, algebraists, and historians of science will find the work a valuable reference. The book may also serve as a supplemental text for courses in abstract algebra or the history of mathematics.

From inside the book

Contents

History of Ring Theory 41
40
History of Field Theory
63
History of Linear Algebra
79
Emmy Noether and the Advent of Abstract Algebra 91
90
A Course in Abstract Algebra Inspired by History
103
Biographies of Selected Mathematicians
113
Index 165
164
18
166
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page 3 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.
Page 162 - Two traits determined above all her nature: First, the native productive power of her mathematical genius. She was not clay, pressed by the artistic hands of God into a harmonious form, but rather a chunk of human primary rock into which he had blown his creative breath of life.
Page 12 - Every equation can have as many distinct roots (values of the unknown quantity) as the number of dimensions of the unknown quantity in the equation.
Page 126 - But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.
Page 127 - V 6 ), which to the best of my knowledge have never been established before. The excessive length that is to be feared in the definitions of the more complicated operations is partly inherent in the nature of the subject but can for the most part be avoided.
Page 162 - Nevertheless she enjoyed the recognition paid her; she could answer with a bashful smile like a young girl to whom one had whispered a compliment. No one could contend that the Graces had stood by her cradle; but if we in Gottingen often chaffingly referred to her as "der Noether...

Bibliographic information