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A History of Abstract Algebra

Israel Kleiner
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We do not plan to review this book.

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