## Exploratory Galois TheoryCombining a concrete perspective with an exploration-based approach, Exploratory Galois Theory develops Galois theory at an entirely undergraduate level. The text grounds the presentation in the concept of algebraic numbers with complex approximations and assumes of its readers only a first course in abstract algebra. For readers with Maple or Mathematica, the text introduces tools for hands-on experimentation with finite extensions of the rational numbers, enabling a familiarity never before available to students of the subject. The text is appropriate for traditional lecture courses, for seminars, or for self-paced independent study by undergraduates and graduate students. |

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### Contents

Preliminaries | 5 |

Algebraic Numbers Field Extensions and Minimal Polynomials | 22 |

Working with Algebraic Numbers Field Extensions and Minimal Polynomials | 39 |

Multiply Generated Fields | 63 |

18 Isomorphisms from Multiply Generated Fields | 78 |

19 Fields and Splitting Fields Generated by Arbitrarily Many Algebraic Numbers | 83 |

20 Exercise Set 1 | 86 |

Maple and Mathematica | 89 |

29 Exercise Set 2 | 149 |

Some Classical Topics | 152 |

31 Cyclic Extensions over Fields with Roots of Unity | 156 |

32 Binomial Equations | 161 |

33 RulerandCompass Constructions | 163 |

34 Solvability by Radicals | 171 |

35 Characteristic p and Arbitrary Fields | 177 |

36 Finite Fields | 186 |

22 Exercise Set 2 | 100 |

The Galois Correspondence | 103 |

24 The Galois Group | 105 |

25 Invariant Polynomials Galois Resolvents and the Discriminant | 115 |

26 Exercise Set 1 | 127 |

27 Distinguishing Numbers Determining Groups | 128 |

Maple and Mathematica | 137 |

Historical Note | 193 |

Subgroups of Symmetric Groups | 197 |

2 The Subgroups of S5 | 198 |

201 | |

205 | |

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### Common terms and phrases

abelian algebraic element algebraic extension algebraic numbers AlgFields arbitrary fields arithmetic combination automorphism chapter coefficients complex numbers conjugate consider constructible numbers contains cyclic declared field Definition denote Details of field Dimension over Q disc(p Division Algorithm equation Euclidean Algorithm evaluation homomorphism Example Exercise exists expressed extension field field L Field Theorem finite field Fix(H function Gal(L/N Galois correspondence Galois group Galois resolvent Galois theory greatest common divisor Hence Hint homomorphism induction intermediate field inverse irreducible polynomial K(ai K(ft K[Xi linear factor minimal polynomial monic monomial monomorphism multiplicity Multiply Generated Fields normal p e K[X permutation polynomial in K[X polynomial p e polynomial ring primitive mh root Proposition Prove quotient radical extension rational numbers reduced form result Root Approximations root of unity separable polynomial splitting field extension subfield subgroup of S4 subgroups of order Suppose symmetric polynomial unique Z/nZ

### Popular passages

Page 204 - Remarques relatives 1 à des classes très-étendues de quamités dom la valeur n'est ni rationnelle ni même réductible à des irrationnelles algébriques; 2 à un passage du livre des Principes où Newton calcule l'action exercée par une sphère sur un poim extérieur et Nouvelle demonstration d'un théorème sur les irrationnelles aigébriques.