## Galois Theory, Third EditionIan Stewart's Galois Theory has been in print for 30 years. Resoundingly popular, it still serves its purpose exceedingly well. Yet mathematics education has changed considerably since 1973, when theory took precedence over examples, and the time has come to bring this presentation in line with more modern approaches. To this end, the story now begins with polynomials over the complex numbers, and the central quest is to understand when such polynomials have solutions that can be expressed by radicals. Reorganization of the material places the concrete before the abstract, thus motivating the general theory, but the substance of the book remains the same. |

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### Contents

Chapter 1 Classical Algebra | 1 |

Chapter 2 The Fundamental Theorem of Algebra | 18 |

Chapter 3 Factorization of Polynomials | 33 |

Chapter 4 Field Extensions | 51 |

Chapter 5 Simple Extensions | 60 |

Chapter 6 The Degree of an Extension | 69 |

Chapter 7 RulerandCompass Constructions | 78 |

Chapter 8 The Idea Behind Galois Theory | 88 |

Chapter 15 Solution by Radicals | 157 |

Chapter 16 Abstract Rings and Fields | 167 |

Chapter 17 Abstract Field Extensions | 180 |

Chapter 18 The General Polynomial | 193 |

Chapter 19 Regular Polygons | 211 |

Chapter 20 Finite Fields | 231 |

Chapter 21 Circle Division | 237 |

Chapter 22 Calculating Galois Groups | 257 |

Chapter 9 Normality and Separability | 111 |

Chapter 10 Counting Principles | 121 |

Chapter 11 Field Automorphisms | 129 |

Chapter 12 The Galois Correspondence | 135 |

Chapter 13 A Worked Example | 139 |

Chapter 14 Solubility and Simplicity | 147 |

Chapter 23 Algebraically Closed Fields | 268 |

Chapter 24 Transcendental Numbers | 276 |

287 | |

290 | |

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### Common terms and phrases

abelian algebraic extension automorphism Chapter coefﬁcients complex numbers construction contains coprime Corollary cubic cyclic deﬁned deﬁnition divides elementary symmetric polynomials example Exercise exists extension L:K ﬁeld extension ﬁeld of characteristic ﬁnd ﬁnite extension ﬁnite ﬁeld ﬁnite group ﬁrst ﬁxed ﬁeld following true formula Fundamental Theorem Galois correspondence Galois group Galois theory Galois’s Gauss group G Hence homomorphism induction integer integral domain intermediate ﬁeld irreducible polynomial isomorphic K-automorphism K-monomorphism Lemma Mark the following mathematical minimal polynomial modulo monic monomorphism multiplicative nonzero normal closure normal extension normal subgroup notation permutations polynomial equation polynomial f polynomial of degree primitive PROOF Let Proposition prove quadratic quartic quintic equation quotient radical extension real numbers ring root of unity Rufﬁni ruler and compasses simple algebraic extensions simple extension soluble by radicals solution speciﬁc splitting ﬁeld square roots subring sufﬁcient Suppose Theorem of Algebra trisected unique zeros