| Lec # | TOPICS | READINGS |
|---|---|---|
| 1 | Course Overview | Elliptic Curves (PDF) |
| 2 | Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition) | Janusz, sections I.1-3. |
| 3 | Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals | Janusz, section I.3. |
| 4 | Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, Discriminant | Janusz, sections I.4-5. |
| 5 | Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable Case | Janusz, sections I.5-6. |
| 6 | Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, Discriminant | Janusz, sections I.6-7. |
| 7 | Discriminants and Ramification, Norms of Ideals | Janusz, sections I.7-8. |
| 8 | Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case) | Janusz, sections I.8 and I.10. |
| 9 | Linearly Disjoint Extensions; Cyclotomic Fields (General Case) | Janusz, sections I.9, I.10, and I.11. See also this supplement (PDF) |
| 10 | Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, Lattices | Janusz, sections I.11, I.12, and I.13. |
| 11 | Lattices and Ideal Classes, Minkowski's Theorem, Finiteness of the Class Group; Dirichlet's Units Theorem | Janusz, sections I.12 and I.13. |
| 12 | Proof of Dirichlet's Units Theorem | Janusz, section I.13. |
| 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) | Janusz, sections II.1-II.3. |
| 14 | In-class Midterm Exam | |
| 15 | Ostrowski's Theorem (cont.); Exponential and Logarithm Series; Hensel's Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values | Janusz, sections II.2 and II.3. |
| 16 | Extension of Nonarchimedean Absolute Values | Janusz, section II.3. |
| 17 | Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified Extensions | Janusz, sections II.3 and II.5. |
| 18 | Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols | Janusz, sections III.1 and III.2. |
| 19 | Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic Case | Janusz, sections III.3 and IV.1. |
| 20 | More on Ray Class Groups; Idelic Interpretation | Janusz, section IV.1. Neukirch, section VI.1. |
| 21 | Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet's Theorem and Generalizations | Janusz, section IV.2. |
| 22 | Chebotarev Density Theorem; Arakelov Class Group | See the Arakelov Class Group Notes by Rene Schoof (PDF) Janusz, section IV.3. |
| 23 | Arakelov Class Group (cont.); Local Class Field Theory | See the Arakelov Class Group Notes by Rene Schoof (PDF) Neukirch, section III. Milne's Notes. |
| 24 | Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem | Neukirch, sections III and VI. Milne's Notes. Cassels-Fröhlich. |
| 25 | Class Field Towers; Complex Multiplication | Cassels-Fröhlich. |