Local Class Field Theory 2025

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.