Lec # | TOPICS | KEY DATES |
---|---|---|

1 | Course Overview | |

2 | Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition) | |

3 | Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals | |

4 | Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, Discriminant | Problem set 1 due |

5 | Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable Case | |

6 | Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, Discriminant | Problem set 2 due |

7 | Discriminants and Ramification, Norms of Ideals | |

8 | Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case) | Problem set 3 due |

9 | Linearly Disjoint Extensions; Cyclotomic Fields (General Case) | |

10 | Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, Lattices | Problem set 4 due |

11 | Lattices and Ideal Classes, Minkowski's Theorem, Finiteness of the Class Group; Dirichlet's Units Theorem | |

12 | Proof of Dirichlet's Units Theorem | Problem set 5 due |

13 | Absolute Values; Completions of Fields with Respect to an Absolute Value, Examples; Dichotomy between Archimedean Nonarchimedean Absolute Values; Absolute Values Coming from Discrete Valuation Rings; Normalized Absolute Values (Places), Statement of the Product Formula for Number Fields; Classification of Completions of the Rational Numbers (Ostrowski's Theorem) | |

14 | In-class Midterm Exam | Problem set 6 due |

15 | Ostrowski's Theorem (cont.); Exponential and Logarithm Series; Hensel's Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values | |

16 | Extension of Nonarchimedean Absolute Values | Detachable midterm exam due Problem set 7 due |

17 | Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified Extensions | Problem set 8 due |

18 | Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols | |

19 | Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic Case | Problem set 9 due |

20 | More on Ray Class Groups; Idelic Interpretation | |

21 | Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet's Theorem and Generalizations | Problem set 10 due |

22 | Chebotarev Density Theorem; Arakelov Class Group | |

23 | Arakelov Class Group (cont.); Local Class Field Theory | |

24 | Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem | |

25 | Class Field Towers; Complex Multiplication | Take-home final exam due |