A History of Abstract Algebra
This book gives an overview of the origin and development of the basic ideas of modern abstract algebra. The author shows how abstract algebra has arisen from the study of solutions of polynomial equations to a theory of abstract algebraic structures and axiomatic systems, such as groups, rings and fields. He briefly describes the transition from “classical” to “modern” theory, which happened in the 19th century, i.e. the process of creation and development of modern structural algebra. He demonstrates that abstract algebra became an independent, interesting and flourishing subject of mathematics in the early decades of the twentieth century thanks to the pioneering work of Emmy Noether. In the first short chapter, the author deals with the prehistory and history of classical algebra. He starts with the early roots of algebra and he goes through Greek algebra, Islamic algebraic accomplishments, the discovery of solutions of cubic and quartic equations, and the Fundamental Theorem of Algebra to the birth of symbolic algebra.
In the second chapter, the book indicates the roots of group theory (for example some of the origins of group theory can be found in linear algebra, number theory, geometry and analysis), illustrates the development of specialized theories of groups (for example permutation groups, Abelian groups and transformation groups) and explains the role of abstraction and consolidation of the abstract group concept and further development and accomplishment of modern group theory. The third chapter deals with the creation and development of ring theory. It starts with non-commutative ring theory (hypercomplex number systems, their classification and their structure) and moves on to commutative ring theory, which is analysed from a mathematical and an historical point of view (the most important events and results on algebraic number theory, algebraic geometry and invariant theory are presented). At the end of the chapter, the abstract definition of a ring is covered and interesting results of Emmy Noether and Emil Artin are offered to the reader in a simple, understandable way.
The fourth chapter is divided into nine parts concentrating on the historical aspects of the birth and development of Galois theory, algebraic number theory (Dedekind’s and Kronecker’s ideas, Hensel’s p-adic numbers, etc.), algebraic geometry (fields of algebraic or rational functions), the role of symbolic algebra and the abstract definition of a field. The fifth chapter contains a description of the evolution of linear algebra and its connections with group theory, ring theory and field theory. Only an overview of the fundamental developing aspects of linear algebra is given (for example problems of solving linear equations, the birth and application of determinants, matrices and linear transformations, and the role of linear independence, basis, dimension and vector spaces in the development of algebra). The sixth chapter is devoted to an analysis of Noether’s fundamental algebraic works on invariant theory, commutative algebra, non-commutative algebra and representation theory. Applications of Noether’s results on non-commutative to commutative algebra are also explained. The seventh chapter offers very interesting suggestions to instructors on how the history of abstract algebra could be integrated into their teaching. In the eighth chapter, there are six biographies of major contributors to the development of modern algebra (A. Cayley, R. Dedekind, E. Galois, C. F. Gauss, W. R. Hamilton and E. Noether) describing their lives and works, which are written as readable mini-essays.
The book is a far from exhaustive account of the history of abstract algebra but for readers who want to pursue the subject in more detail, the author indicates where additional material can be found. In each chapter, the author makes extensive references to relevant literature. The book can be recommended to mathematicians, teachers of mathematics (especially of algebra), historians of the sciences and students, who can find many useful references and ideas for their research, teaching or studies. The book may also serve as a supplementary text for courses on the history of modern mathematics or abstract algebra.