## The Beginnings and Evolution of AlgebraThe elements of algebra were known to the ancient Mesopotamians at least 4000 years ago. Today algebra stands as one of the cornerstones of modern mathematics. How then did the subject evolve? How did its constituent ideas and concepts arise, and how have they changed over the years? These are the questions that the authors address in this work. The authors challenge the existing view that the development of algebra was driven by the investigation of determinate equations and in particular their solution by radicals. In short they claim that the study of indeterminate equations was no less important. Historians of mathematics, as well as working algebraists who want to look into the history of their subject, will find this an illuminating read. |

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### Contents

Chapter | 11 |

Chapter 3 | 35 |

Chapter 4 | 49 |

The first advances in algebra in Europe | 55 |

Algebraic symbolism in Europe The German cossists and the development of algebra in Italy | 59 |

Chapter 5 | 67 |

The Algebra of Rafael Bombelli Introduction of complex numbers | 71 |

François Viète | 75 |

The fundamental theorem of algebra | 94 |

Gauss criticism | 98 |

The problem of solution of equations by radicals | 100 |

Proof of the unsolvability of the general quintic by radicals | 102 |

Chapter 7 | 109 |

Equations with an Abelian group | 114 |

Galois theory | 115 |

The evolution of group theory in the 19th century | 120 |

Creation of a literal calculus | 77 |

Genesis triangulorum | 80 |

Indeterminate equations in the work of Viète | 85 |

Beginning of the theory of determinate equations | 87 |

Chapter 6 | 91 |

Descartes treatment of determinate equations | 93 |

The victorious march of group theory | 125 |

Chapter 8 | 129 |

Chapter 9 | 149 |

Conclusion | 161 |

References | 175 |

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### Common terms and phrases

addition ancient applied Arabic arbitrary arithmetic associated Babylonians beginning Book calculus called century classes coefficients commutative complex complex numbers composition connection considered construction contains corresponding cubic curve defined denoted determinate Diophantus discovery divisible domain elements equal equation equivalent Euler evolution example expressed extension factors field Figure formula functions fundamental Galois Gauss geometric geometric algebra given ideal important integers introduced known later letter linear literal magnitudes mathematicians mathematics means method multiplication namely negative numbers obtained operations period permutations polynomial positive possible powers prime problem proof propositions proved quadratic equations radicals rational reduced respect result ring roots rules segments showed sides solution solvable solve square subgroup substitution symbols takes theorem theory transformation triangle unknown values Viète wrote