Complete course contents

All twenty-four lessons and ninety-six exercise solutions are available. Read the study guide or the external reading guide.

  1. Profinite groups and infinite Galois theory
  2. Cohomology of cyclic groups and the Herbrand quotient
  3. Hilbert's Theorem 90 and Kummer theory
  4. Frobenius lifts and abstract reciprocity
  5. The reciprocity law and the class field correspondence
  6. Local reciprocity and norm groups
  7. Formal groups and Lubin–Tate modules
  8. Lubin–Tate division fields
  9. Explicit local reciprocity and the existence theorem
  10. Abelian ramification, conductors and Hasse–Arf
  11. Hilbert symbols and local conics
  12. Weil groups and one-dimensional representations
  13. Idèles in extensions and their cohomology
  14. The Herbrand quotient of the idèle class group
  15. The norm index bound and Hasse's norm theorem
  16. The global reciprocity law
  17. Global existence and the idèlic class field correspondence
  18. Ray class fields, conductors and ideal reciprocity
  19. Hilbert and ring class fields, and quadratic prime forms
  20. Kronecker–Weber and the maximal abelian extension of the rationals
  21. Artin L-functions, conductors and discriminants
  22. The Chebotarev density theorem
  23. Power residue symbols and reciprocity laws
  24. Brauer groups of local and global fields

The optional higher-dimensional local epsilon existence and local orthogonal formula retain the precise qualifications described in lesson 12 and lesson 21.