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This course continues the study of number fields begun in the course Number Theory I of the previous semester. Topics include p-adic numbers, local fields, valuations, Galois theory of valuations, ramification theory, and some local and global abelian class field theory.
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 tba. 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 report mistakes and suggest improvements to Prof. Pink.
| Date | Notes | Contents |
|---|---|---|
| Wednesday, February 21 | §8.1 | p-adic numbers( Maple Worksheet) |
| Friday, February 23 | §8.1-8.2 | p-adic numbers, valuations |
| Wednesday, February 28 | §8.3 | complete valuations |
| Monday, March 4 | §8.4 | absolute values |
| Wednesday, March 6 | §8.5-8.6 | completion of a metric space, complete absolute values |
| Friday, March 8 | §8.7 | power series |
| Wednesday, March 13 | §9.1-9.2 | normed vector spaces, extensions of complete absolute values |
| Friday, March 15 | §9.2-9.3 | extensions of complete absolute values, Newton polygons |
| Wednesday, March 20 | §9.4-9.5 | lifting prime ideals, extensions of absolute values |
| Friday, March 22 | §9.5a §9.5b | extensions of absolute values |
| Wednesday, March 27 | §9.5-6 | extensions of absolute values, local and global fields |
| Wednesday, April 10 | §10.1-10.3 | Topological group, profinite groups, infinite Galois theory |
| Friday, April 12 | §11.1 | Multiplicative group |
| Wednesday, April 17 | §11.2-11.3 | Unframified extensions, tame extensions |
| Friday, April 19 | §11.3 | Tame extensions |
| Wednesday, April 24 | §11.4-11.5 | The lower and upper numbering filtration |
| Friday, April 26 | §11.5 | The upper numbering filtration |
| Friday, May 03 | §11.6, 10.4 | Kummer theory, Abelian extensions of Qp |
| Wednesday, May 8 | §11.6, 11.7 | Abelian extensions of Qp, the Kronecker-Weber theorem |
| Friday, May 10 | §12.1-12.2 | Cohomology of cyclic groups, some Galois cohomology |
| Wednesday, May 15 | §12.3 | The reciprocity isomorphism |
| Friday, May 17 | §12.3 | The reciprocity isomorphism |
| Wednesday, May 22 | §12.4-13.2 | The existence theorem, Ideles, Idele classes |
| Friday, May 24 | §13.3 | The reciprocity isomorphism |
| Wednesday, May 29 | §13.4-13.5 | Class field, Reciprocity laws |