## Galois TheoryIn the nineteenth century, French mathematician Evariste Galois developed the Galois theory of groups-one of the most penetrating concepts in modem mathematics. The elements of the theory are clearly presented in this second, revised edition of a volume of lectures delivered by noted mathematician Emil Artin. The book has been edited by Dr. Arthur N. Milgram, who has also supplemented the work with a Section on Applications. |

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### Contents

LINEAR ALGEBRA | 1 |

C Homogeneous Linear Equations | 2 |

D Dependence and Independence of Vectors | 4 |

E Nonhomogeneous Linear Equations | 9 |

F Determinants | 11 |

FIELD THEORY | 21 |

B Polynomials | 22 |

C Algebraic Elements | 25 |

J Roots of Unity | 56 |

K Noether Equations | 57 |

L Kummers Fields | 59 |

M Simple Extensions | 64 |

N Existence of a Normal Basis | 66 |

O Theorem on Natural Irrationalities | 67 |

APPLICATIONS | 69 |

B Permutation Groups | 70 |

D Splitting Fields | 30 |

E Unique Decomposition of Polynomials into Irreducible Factors | 33 |

F Group Characters | 34 |

G Applications and Examples to Theorem 13 | 38 |

H Normal Extensions | 41 |

I Finite Fields | 49 |

C Solution of Equations by Radicals | 72 |

D The General Equation of Degree n | 74 |

E Solvable Equations of Prime Degree | 76 |

F Ruler and Compass Constructions | 80 |