## Algebra: Volume I: Fields and Galois TheoryFrom Math Reviews: "This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the framework of the development of carefully selected problems. Thus, for instance, the transformation of the classical geometrical problems on constructions with ruler and compass in their algebraic setting in the first chapter introduces the reader spontaneously to such fundamental algebraic notions as field extension, the degree of an extension, etc... The book ends with an appendix containing exercises and notes on the previous parts of the book. However, brief historical comments and suggestions for further reading are also scattered through the text." |

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### Contents

3 | |

Algebraic Extensions | 15 |

Simple Extensions | 21 |

Fundamentals of Divisibility | 33 |

Prime Factorization in Polynomial Rings Gausss Theorem 45 | 44 |

Polynomial Splitting Fields | 55 |

Separable Extensions | 65 |

Galois Extensions 75 | 74 |

Norm and Trace | 133 |

Binomial Equations 143 | 142 |

Solvability of Equations | 165 |

Integral Ring Extensions | 191 |

The Transcendence of | 203 |

Transcendental Field Extensions | 209 |

Hilberts Nullstellensatz | 217 |

Problems and Remarks | 231 |

Finite Fields Cyclic Groups and Roots of Unity | 83 |

Group Actions | 93 |

Applications of Galois Theory to Cyclotomic Fields | 103 |

Further Steps into Galois Theory 115 | 114 |

283 | |

287 | |