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A History of Abstract Algebra by Israel Kleiner

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A History of Abstract Algebra written by Israel Kleiner, Department of Mathematics and Statistics, York University, Toronto, Canada, kleiner@rogers.com. 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 of this book: * 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.

A History of Abstract Algebra written by Israel Kleiner cover the following topics.

  • Preface

  • 1. History of Classical Algebrav
    1.1 Early roots
    1.2 The Greeks
    1.3 Al-Khwarizmi
    1.4 Cubic and quartic equations
    1.5 The cubic and complex numbers
    1.6 Algebraic notation: Viète and Descartes
    1.7 The theory of equations and the Fundamental Theorem of Algebra
    1.8 Symbolical algebra

  • 2. History of Group Theory
    2.1 Sources of group theory
    2.1.1 Classical Algebra
    2.1.2 Number Theory
    2.1.3 Geometry
    2.1.4 Analysis
    2.2 Development of “specialized” theories of groups
    2.2.1 Permutation Groups
    2.2.2 Abelian Groups
    2.2.3 Transformation Groups
    2.3 Emergence of abstraction in group theory
    2.4 Consolidation of the abstract group concept; dawn of abstract group theory
    2.5 Divergence of developments in group theory

  • 3. History of Ring Theory
    3.1 Noncommutative ring theory
    3.1.1 Examples of Hypercomplex Number Systems
    3.1.2 Classification
    3.1.3 Structure
    3.2 Commutative ring theory
    3.2.1 Algebraic Number Theory
    3.2.2 Algebraic Geometry
    3.2.3 Invariant Theory
    3.3 The abstract definition of a ring
    3.4 Emmy Noether and Emil Artin
    3.5 Epilogue

  • 4. History of Field Theory
    4.1 Galois theory
    4.2 Algebraic number theory
    4.2.1 Dedekind’s ideas.
    4.2.2 Kronecker’s ideas
    4.2.3 Dedekind vs Kronecker
    4.3 Algebraic geometry
    4.3.1 Fields of Algebraic Functions
    4.3.2 Fields of Rational Functions
    4.4 Congruences
    4.5 Symbolical algebra
    4.6 The abstract definition of a field
    4.7 Hensel’s p-adic numbers
    4.8 Steinitz
    4.9 A glance ahead

  • 5. History of Linear Algebra
    5.1 Linear equations
    5.2 Determinants
    5.3 Matrices and linear transformations
    5.4 Linear independence, basis, and dimension
    5.5 Vector spaces

  • 6 Emmy Noether and the Advent of Abstract Algebra
    6.1 Invariant theory
    6.2 Commutative algebra
    6.3 Noncommutative algebra and representation theory
    6.4 Applications of noncommutative to commutative algebra
    6.5 Noether’s legacy

  • 7. A Course in Abstract Algebra Inspired by History
    Problem I: Why is (-1)(-1) = 1?
    Problem II: What are the integer solutions of x2 + 2 = y3?
    Problem III: Can we trisect a 60? angle using only straightedge and compass?
    Problem IV: Can we solve x5 - 6x + 3 = 0 by radicals?
    Problem V: “Papa, can you multiply triples?”
    General remarks on the course

  • 8. Biographies of Selected Mathematicians
    8.1 Arthur Cayley (1821–1895)
    8.1.1 Invariants
    8.1.2 Groups
    8.1.3 Matrices
    8.1.4 Geometry
    8.1.5 Conclusion
    8.2 Richard Dedekind (1831–1916)
    8.2.1 Algebraic Numbers
    8.2.2 Real Numbers
    8.2.3 Natural Numbers
    8.2.4 OtherWorks
    8.2.5 Conclusion
    8.3 Evariste Galois (1811–1832)
    8.3.1 Mathematics
    8.3.2 Politics .
    8.3.3 The duel
    8.3.4 Testament
    8.3.5 Conclusion
    8.4 Carl Friedrich Gauss (1777–1855)
    8.4.1 Number theory
    8.4.2 Differential Geometry, Probability, and Statistics
    8.4.3 The diary
    8.4.4 Conclusion
    8.5 William Rowan Hamilton (1805–1865)
    8.5.1 Optics
    8.5.2 Dynamics
    8.5.3 Complex Numbers
    8.5.4 Foundations of Algebra
    8.5.5 Quaternions
    8.5.6 Conclusion
    8.6 Emmy Noether (1882–1935)
    8.6.1 Early Years
    8.6.2 University Studies
    8.6.3 Göttingen
    8.6.4 Noether as a Teacher
    8.6.5 Bryn Mawr
    8.6.6 Conclusion

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