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Number theory

Class field theory

Reciprocity for local and global fields, built from Galois theory, cyclic cohomology and radicals, then made explicit with Lubin–Tate groups, idèles and ray class fields.

Take first

Begin with finite Galois theory and basic topology. Number fields, local fields and analytic number theory supply the arithmetic prerequisites for later lessons. Each lesson identifies the particular results it uses.

Reading paths

Lesson numbers remain stable. These paths select the sections needed for a particular question. Read the full study guide and the proofs and freely readable references.

From concrete norms to local reciprocity

Start with automorphisms and norm-one elements; inspect the local cyclic calculation before constructing the abstract map.

  1. Lesson 1: all
  2. Lesson 3: all
  3. Lesson 2: all
  4. Lesson 6: 1–3: arithmetic data and cyclic axiom
  5. Lesson 4: all: norm-class construction
  6. Lesson 5: all: isomorphism and correspondence
  7. Lesson 6: 4–6: arithmetic reciprocity and norms
  8. Lesson 7: all
  9. Lesson 8: all
  10. Lesson 9: all
  11. Lesson 10: all
  12. Lesson 11: all
  13. Lesson 12: 1–6: local Weil groups and factors

From local obstructions to global classes

See a complete constant-field calculation, then prove its finite-place and lattice mechanism before proving principal-symbol cancellation.

  1. Lesson 13: all
  2. Lesson 14: 1–2: explicit calculation and three modules
  3. Lesson 14: 3–8: general proof and splitting applications
  4. Lesson 15: 1–8: norm bound and cyclic axiom
  5. Lesson 16: 1: cyclotomic and constant-field cancellation
  6. Lesson 16: 2–8: map on classes and local comparison
  7. Lesson 17: all: existence and function-field Weil groups
  8. Lesson 18: all: ray conditions

Applications of reciprocity

Choose the arithmetic question after the local and global core.

  1. Lesson 20: rational ray fields and cyclotomic action
  2. Lesson 19: Hilbert and ring class fields, quadratic prime forms
  3. Lesson 23: power symbols and primary reciprocity laws
  4. Lesson 21: Artin factors and functional equations
  5. Lesson 22: Dirichlet density and ray primes
  6. Lesson 22: 6A: full Hasse–Minkowski after the ray-prime theorem

Cohomology, Weil extensions and towers

Follow the final proofs back to the earlier applications they complete.

  1. Lesson 24: all: Brauer invariants, fundamental classes, Weil extensions and tower bounds
  2. Lesson 12: 7: global compatibility, with lesson 17 in function fields and lesson 24 in number fields
  3. Lesson 19: 9: infinite iteration; the infinite-iteration; the infinite-tower theorem is proved in lesson 24 theorem is proved in lesson 24s, with the bound proved in lesson 24
  4. Lesson 21: 8: global orthogonal consequence, with its precisely located programme prerequisites

Lessons

  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

Read and edit

This HTML edition contains all 24 lessons. The complete LaTeX file contains all their text; the ZIP supplies the figures, original lesson files and reproduction instructions.

  1. Complete LaTeX edition
  2. Complete editable source ZIP

Authors and status

Written by GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort). Self-checked by the writing AI, GPT-6.1 Sol, Ultra. No independent review is claimed.

Twenty-four lessons and ninety-six exercise solutions are supplied. Consult the prerequisite record for full proof availability.

References

Each lesson ends with its own list of references.

The reading edition supplies the complete course LaTeX and complete editable source ZIP. Each lesson supplies its individual LaTeX and Markdown source.

Licence

Original course text, figures and code. It is dedicated under CC0 1.0: no copyright is claimed, and to the extent any right exists anywhere, it is waived. Rendered DejaVu font glyphs retain the DejaVu font licence. Cited mathematical works retain their own rights; their source files are outside the archive. Font notices: DejaVu, STIX, BaKoMa. Works cited in the lessons keep their own licences.

Licensing on this site · LICENSING.md

The Markdown files of this course are in the editable source ZIP.