## Galois TheorySince 1973, Galois Theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fourth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today's algebra students. New to the Fourth EditionThe replacement of the topological proof of the fundame |

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### Contents

Historical Introduction | 1 |

1 Classical Algebra | 17 |

2 The Fundamental Theorem of Algebra | 35 |

3 Factorisation of Polynomials | 47 |

4 Field Extensions | 63 |

5 Simple Extensions | 71 |

6 The Degree of an Extension | 79 |

7 RulerandCompass Constructions | 87 |

14 Solubility and Simplicity | 161 |

15 Solution by Radicals | 171 |

16 Abstract Rings and Fields | 181 |

17 Abstract Field Extensions | 193 |

18 The General Polynomial Equation | 205 |

19 Finite Fields | 221 |

20 Regular Polygons | 227 |

21 Circle Division | 243 |

8 The Idea Behind Galois Theory | 107 |

9 Normality and Separability | 129 |

10 Counting Principles | 137 |

11 Field Automorphisms | 145 |

12 The Galois Correspondence | 151 |

13 A Worked Example | 155 |

22 Calculating Galois Groups | 267 |

23 Algebraically Closed Fields | 277 |

24 Transcendental Numbers | 285 |

25 What Did Galois Do or Know? | 295 |

309 | |

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### Common terms and phrases

abelian algebraic angle assume automorphism called Chapter characteristic circle claim clearly coefficients complex numbers condition consider construction contains contradiction Corollary correspondence cubic cyclic defined Definition degree distinct divides divisible elements equal equation example Exercise exists expression extension fact factors field extension Figure finite fixed formula functions Galois group give given Hence idea identity implies independent induction integer irreducible irreducible polynomial isomorphic least Lemma linear mathematical method minimal polynomial monomorphism multiple namely non-zero normal notation particular permutations possible prime primitive problem Proof properties Proposition prove quadratic quintic radicals rational reducible regular relation result ring root of unity separable simple solution solved splitting field square roots subfield Suppose symmetric Theorem transcendental true unique write zeros