Indian Institute of Technology, Bombay

Powai, Mumbai 400076 India

1 | Field Extensions | 6 |
---|---|---|

1.1 Basic Facts | 6 | |

1.2 Basic Examples | 9 | |

1.3 Norm, Trace and Discriminant | 12 | |

2 | Ring Extensions | 15 |

2.1 Basic Processes in Ring Theory | 15 | |

2.2 Noetherian Rings and Modules | 17 | |

2.3 Integral Extensions | 19 | |

2.4 Discriminant of a Number Field | 21 | |

3 | Dedekind Domains and Ramification Theory | 26 |

3.1 Dedekind Domains | 27 | |

3.2 Extensions of Primes | 32 | |

3.3 Kummer's Theorem | 35 | |

3.4 Dedekind's Discriminant Theorem | 37 | |

3.5 Ramification in Galois Extensions | 38 | |

3.6 Decomposition and Inertia Groups | 40 | |

3.7 Quadratic and Cyclotomic Extensions | 42 | |

4 | Class Number and Lattices | 46 |

4.1 Norm of an ideal | 46 | |

4.2 Embeddings and Lattices | 48 | |

4.3 Minkowski's Theorem | 52 | |

4.4 Finiteness of Class Number and Ramification | 53 | |

Bibliography | 56 | |

A | Appendix: Notes on Galois Theory | 57 |

A.1 Preamble | 57 | |

A.2 Field Extensions | 58 | |

A.3 Splitting Fields and Normal Extensions | 60 | |

A.4 Separable Extensions | 62 | |

A.5 Galois Theory | 63 | |

A.6 Norms and Traces | 67 | |

B | Appendix: Discriminants in Algebra and Arithmetic | 70 |

B.1 Discriminant in High School Algebra | 70 | |

B.2 Discriminant in College Algebra | 74 | |

B.3 Discriminant in Arithmetic | 77 | |

References | 82 | |

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