Información General

• Autores: Israel Kleiner
• Editores: Boston : Birkhäuser, cop. 2007
• Año de publicación: 2007
• País: Estados Unidos
• Idioma: inglés
• ISBN: 978-0-8176-4684-4, 978-0-8176-4685-1
• Texto completo no disponible (Saber más ...)

Resumen

• * Excellent reference or supplemental text for a first course in abstract algebra * Provides historical context from which the reader may gain a deeper appreciation of the subject * 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 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 intellectual lineage behind many 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.

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Índice

• Preface.-Chapter 1: Classical Algebra.-Early roots.-The Greeks.-Al-Khwarizmi.-Cubic and quartic equations.-The cubic and complex numbers.-Algebraic notation: Viète and Descartes.-The theory of equations and the Fundamental Theorem of Algebra.-Symbolical algebra.-References.-Chapter 2: Group Theory.-Sources of group theory.- Development of "specialized" theories of groups.-Emergence of abstraction in group theory.-Consolidation of the abstract group concept; dawn of abstract group theory.-Divergence of developments in group theory.-References.-Chapter 3: Ring Theory.- Noncommutative ring theory.-Commutative ring theory.-The abstract definition of a ring.-Emmy Noether and Emil Artin.-Epilogue.-References.-Chapter 4: Field Theory.- Galois theory.-Algebraic number theory.-Algebraic geometry.-Symbolical algebra.- The abstract definition of a field.-Hensel’s p-adic numbers.-Steinitz.-A glance ahead.- References.-Chapter 5: Linear Algebra.-Linear equations.-Determinants Matrices and linear transformations.-Linear independence, basis, and dimension.- Vector spaces.-References.-Chapter 6: Emmy Noether and the Advent of Abstract Algebra.-Invariant theory.-Commutative algebra.-Noncommutative algebra and representation theory.-Applications of noncommutative to commutative algebra.- Noether’s legacy.-References.-Chapter 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 600 angle using only straightedge and compass? .-Problem IV: Can we solve x5 - 6x + 3 = 0? .-Problem V: "Papa, can you multiply triples?" .-General remarks on the course.-References.-Chapter 8: Biographies of Selected Mathematicians.-Cayley.-Invariants.-Groups.-Matrices.-Geometry.-Conclusion.-References.-Dedekind.-Algebraic numbers.-Real numbers.-Natural numbers.-Other works.-Conclusion.-References.-Galois.-Mathematics.-Politics.- The duel.-Testament.-Conclusion.-References.-Gauss.-Number theory.-Differential geometry, probability, statistics.-The diary.-Conclusion.-References.-Hamilton.-Optics.- Dynamics.-Complex numbers.-Foundations of algebra.-Quaternions.-Conclusion.-References.-Noether.-Early years.-University studies.-Göttingen.-Noether as a teacher.- Bryn Mawr.-Conclusion.-References.-Index.-Acknowledgments