| 1 |
Course Overview |
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| 2 |
Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition) |
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| 3 |
Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals |
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| 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 |
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| 6 |
Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, Discriminant |
Problem set 2 due |
| 7 |
Discriminants and Ramification, Norms of Ideals |
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| 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) |
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| 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 |
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| 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) |
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| 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 |
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| 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 |
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| 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 |
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| 21 |
Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet's Theorem and Generalizations |
Problem set 10 due |
| 22 |
Chebotarev Density Theorem; Arakelov Class Group |
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| 23 |
Arakelov Class Group (cont.); Local Class Field Theory |
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| 24 |
Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem |
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| 25 |
Class Field Towers; Complex Multiplication |
Take-home final exam due |