Local Class Field Theory 2025

The lectures will be weekly on Tuesdays from 12:15 to roughly 14:00.

RETAKE: planned for the week of September 8th to September 12th.

Weekly Schedule

• Week 1, April 8th: Introduction, valued fields, profinite groups.
  Recommended exercises: 1.1, 1.2, 1.3, 1.5.
  Optional exercises: 1.4.

• Week 2, April 15th: Local fields, Hensel's lemma.
  Recommended exercises:  1.6, 1.7, 1.10, 1.12, 1.13, 1.14, 1.15.
  Optional exercises: 1.8, 1.9, 1.11.

• Week 3, April 22nd: Extensions of local fields, unramified extensions.
  Recommended exercises: 1.17, 1.18, 1.19.
  Optional exercises: 1.16.

• Week 4, April 29th: Tamely ramified extensions, ramification groups.
  Recommended exercises: 1.20, 1.21, 1.22, 1.24, 1.25, 1.26.
  Optional exercises: 1.23, 1.27, 1.28.

• Week 5, May 6th: Homological algebra, group cohomology.
  Recommended exercises: 2.4, 2.5, 2.6.
  Optional exercises: 2.1, 2.2, 2.3.

• Week 6, May 13th: Cohomology of finite cyclic groups, (co)restriction.
  Recommended exercises: 2.7, 2.8, 2.10, 2.11.
  Optional exercises: 2.9, 2.12.

• Week 7, May 20th: Inflation, cohomology of profinite groups, homology.
  Recommended exercises: 2.13, 2.14, 2.17, 2.18.
  Optional exercises: 2.15, 2.16.

• Week 8, May 27th: More homology, Tate groups, cup products.
  Recommended exercises: 2.19,  2.20, 2.23.
  Optional exercises: 2.21, 2.22.

• Week 9, June 3rd: Cup products, Hilbert 90, Brauer group.
  Recommended exercises: 2.24, 3.2.
  Optional exercises: 2.25, 3.1.

• Week 10, June 10th: No class (break)
  Recommended exercises: X
  Optional exercises: X

• Week 11, June 17th: The Brauer group of a local field.
  Recommended exercises: 3.3, 3.5.
  Optional exercises: 3.4.

• Week 12, June 27th: Constructing the local reciprocity map.
  Recommended exercises: 3.6, 3.8.
  Optional exercises: 3.7.

• Week 13, July 1st: No class (I'm away)
  Recommended exercises: X
  Optional exercises: X

• Week 14, July 8th: Properties of the reciprocity map, Hilbert symbol.
  Recommended exercises: 3.9, 3.11, 3.12.
  Optional exercises: 3.10, 3.13.

• Week 15, July 15th: Hilbert symbol, existence theorem.
  Recommended exercises: 3.15, 3.16, 3.17.
  Optional exercises: 3.14.

I wrote a very brief introduction into the world of university level mathematics olympiads. In terms of prestige they may not quite rival those aimed at high school students, but this does not take away from the enjoyment of toying with some slightly more advanced mathematical notions in a fresh and creative way. 

It is the author’s hope that the reader will find enjoyment in solving these problems, and that at least once in their mathematical career, one of these tricks may prove useful beyond the scope of mere olympiads. In addition, the author hopes that these notes will grow out to become a useful tool for those new to the world of university level olympiads, and that these competitions will bring them as much joy as they have brought yours truly.

You can find the document here.