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This course gives an introduction to the theory of number fields, which are fundamental objects in algebraic number theory. Specific topics are: Algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, Dirichlet L-series, zeta function, prime number theorem (+ other material if time permits). Prerequisites are the material on rings, fields, and Galois theory from the courses Algebra I and II (lecture summary in German here)
The new exercise sheet will usually be posted here on Thursdays. In the exercise session, we will discuss the exercise sheet of the prior week.
The lecture course will be recorded but not live streamed. The recordings are here. The screen notes of the lectures are accessible in the list below. There will be a summary of the whole lecture course containing all definitions and theorems but no explanations or proofs. Please send mistakes and improvements to Prof. Pink.
| Date | Notes | Contents |
|---|---|---|
| Wednesday, September 20 | §1.1-1.2 | Integral ring extensions, prime ideals |
| Friday, September 22 | §1.3-1.7 | Normalization, localization, field extensions, norm and trace, discriminant |
| Wednesday, September 27 | §1.7-1.10 | Discriminant, linearly disjoint extensions, Dedekind rings, fractional ideals |
| Friday, September 29 | §1.10 | Discriminant, linearly disjoint extensions, Dedekind rings, fractional ideals |
| Wednesday, October 4 | §1.10-2.1 | Fractional ideals, ideals, ideal class group, lattices |
| Friday, October 6 | §2.1-2.3 | Lattices, volume, lattice point theorem |
| Wednesday, October 11 | §3.1-3.5 | Number fields, absolute discriminant, absolute norm, real and complex embeddings, quadratic number fields |
| Friday, October 13 | §3.5-3.6 | Quadratic number fields, cyclotomic fields |
| Wednesday, October 18 | §3.6-3.7 | Cyclotomic fields, Quadratic Reciprocity |
| Friday, October 20 | §3.7-4.2 | Quadratic Reciprocity, Euclidean embedding, lattice bounds |
| Wednesday, October 25 | §4.2-4.4 | Lattice bounds, finiteness of the class group, discriminant bounds |
| Friday, October 27 | §4.4 | Discriminant bounds |
| Wednesday, November 1 | §5.1-5.4 | Roots of unity, units, Dirichlet’s unit theorem, the real quadratic case |
| Friday, November 3 | §5.3 | Dirichlet’s unit theorem |
| Wednesday, November 8 | §6.1-6.2 | Modules over Dedekind rings, decomposition of prime ideals |
| Friday, November 10 | §6.2-6.3 | Decomposition of prime ideals, decomposition group |
| Wednesday, November 15 | §6.4-6.5 | Inertia group, Frobenius |
| Friday, November 17 | §6.6 | Relative norm |
| Wednesday, November 22 | §6.6-6.7 | Relative norm, different |
| Friday, November 24 | §6.7-6.8 | Different, relative discriminant |
| Wednesday, November 29 | §6.8,7.1-7.3 | Relative discriminant, Riemann zeta function, Dedekind zeta function, analytic class number formula |
| Friday, December 01 | §7.2 | Dedekind zeta function |
| Wednesday, December 06 | §7.2 | Dedekind zeta function |
| Friday, December 08 | §7.4 | Dirichlet density |
| Wednesday, December 13 | §7.5-7.6 | Primes of absolute degree 1, Dirichlet L-series |
| Friday, December 15 | §7.6-7.7 | Dirichlet L-series, Primes in arithmetic Progression |
| Wednesday, December 20 | §7.8-7.9 | Bonus material: Abelian Artin L-functions, Cebotarev density theorem |
| Friday, December 22 | §7.9 | Bonus material: Cebotarev density theorem |
You can find the blank slides to fill in your own notes during the lectures here.