Proofs and freely accessible references
The course supplies its lesson arguments and exercise solutions; the two higher-dimensional local prerequisites recorded below still lack complete programme proofs. Each external reading below is freely accessible; it supplies material for understanding an argument and never replaces a programme proof. The exact prerequisite record identifies separately owned results and their proof availability. The study guide gives section-level reading paths.
The course uses arithmetic Frobenius. Geometric Frobenius acts by its inverse; the local and global normalizations are compared explicitly in lessons 12, 20 and 21. Both number fields and function fields retain their stated hypotheses and separate arguments.
Free reading editions
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Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
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Teruyoshi Yoshida, Local class field theory via Lubin–Tate theory, arXiv:math/0606108v2.
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Bjorn Poonen, Tate’s Thesis, MIT 18.786 lecture notes (2015).
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John Tate, Number theoretic background (1979), freely available paper.
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Wen-Wei Li, Yanqi Lake Lectures on Algebra: Part 1, author edition dated 2026-06-09.
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Mikhail Ershov, Golod–Shafarevich groups: a survey, author preprint (2012).
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Pierre Deligne, Les constantes des équations fonctionnelles des fonctions L (1973), IAS archive.
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The Stacks project, independence, Kummer and Artin–Schreier sections.
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Andrew Sutherland, The CM torsor, MIT 18.783 Lecture17 (2023).
Proof paths by lesson
1. Profinite groups and infinite Galois theory
Read the lesson. The finite automorphism lemma, finite Galois correspondence, primitive element theorem and integer and polynomial remainder lemmas are proved in section 0. Sections 1–3 prove compactness, embedding extension and the infinite correspondence; sections 4–5 treat Frobenius and the procyclic examples.
Free comparisons: J. S. Milne, Fields and Galois Theory, version 5.10; Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
2. Cohomology of cyclic groups and the Herbrand quotient
Read the lesson. Section 1 proves the exact cyclic integral resolution. Sections 2–4 prove the connecting maps, the complete six-term sequence, multiplicativity, induced-module calculations and lattice invariance.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; Jürgen Neukirch, Class Field Theory — The Bonn Lectures, Online Edition 2.0 (May 2015), edited by Alexander Schmidt.
3. Hilbert's Theorem 90 and Kummer theory
Read the lesson. Sections 1–3 prove independence and both forms of Hilbert 90 in every characteristic. The finite character lemma proves dual size and separation before the full finite and infinite Kummer and Artin–Schreier correspondences.
Free comparisons: J. S. Milne, Fields and Galois Theory, version 5.10; The Stacks project, independence, Kummer and Artin–Schreier sections.
4. Frobenius lifts and abstract reciprocity
Read the lesson. Sections 2–3 give the full abstract degree and valuation hypotheses, including the general permitted value group. Sections 5–7 prove unit descent, independence, composition and all three functorialities. The central finite unramified tower used in descent is constructed explicitly.
Free comparisons: Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
5. The reciprocity law and the class field correspondence
Read the lesson. Sections 1–4 prove the cyclic, abelian, solvable and Sylow steps without assuming a finite norm quotient prematurely. Theorem 5.4 proves norm limitation for every finite separable extension, using its maximal abelian subextension. Sections 5–6 prove the norm-group correspondence and its topology.
Free comparisons: Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
6. Local reciprocity and norm groups
Read the lesson. Sections 1–3 verify both cyclic axioms by a proved normal basis and a complete filtered lifting lemma. This argument includes equal characteristic; it does not use an exponential or logarithm there. Sections 4–5 give reciprocity, norm limitation and openness.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; Jürgen Neukirch, Class Field Theory — The Bonn Lectures, Online Edition 2.0 (May 2015), edited by Alexander Schmidt.
7. Formal groups and Lubin–Tate modules
Read the lesson. Sections 1–3 prove inversion, coefficient correction, every formal-group identity and scalar compatibility. The first nonlinear coefficients are computed explicitly. Sections 4–6 prove the Frobenius equations and the twisted construction over the completed unramified ring.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; Teruyoshi Yoshida, Local class field theory via Lubin–Tate theory, arXiv:math/0606108v2.
8. Lubin–Tate division fields
Read the lesson. Sections 1–2 prove root counts, the division module, independence of the defining series and the full Galois group. The valuation calculation in Theorem 8.3 proves the degree and total ramification directly. The norm sign and the whole tower are retained.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; Teruyoshi Yoshida, Local class field theory via Lubin–Tate theory, arXiv:math/0606108v2.
9. Explicit local reciprocity and the existence theorem
Read the lesson. The completed-ring Frobenius calculation and descent back to algebraic fields give the explicit unit action. The remaining sections prove local existence and the maximal abelian extension, with the sign conventions kept explicit.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; Teruyoshi Yoshida, Local class field theory via Lubin–Tate theory, arXiv:math/0606108v2.
10. Abelian ramification, conductors and Hasse–Arf
Read the lesson. The division-point motion calculation determines all real-index ramification groups. The quotient filtration gives Hasse–Arf for every finite abelian local extension. The conductor, different and character-sum arguments retain their exact local-field prerequisites.
Free comparisons: Teruyoshi Yoshida, Local class field theory via Lubin–Tate theory, arXiv:math/0606108v2; Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
11. Hilbert symbols and local conics
Read the lesson. The norm characterization and finite perfect pairing lead to the tame, real and dyadic quadratic formulas. The product-algebra proof handles local conics, including the split case. All displayed symbol normalizations are proved in the lesson.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03.
12. Weil groups and one-dimensional representations
Read the lesson. The local Weil construction and one-dimensional formulas use the written rank-one Fourier prerequisites. Global Weil constructions are supplied in lessons 17 and 24. General higher-dimensional epsilon existence is supplied by LG-GAL-06, Theorem 3.0 and sections 3A–3D; its uniqueness theorem is Theorem 3.2. These are programme proofs, distinct from the external reading sources.
Free comparisons: Bjorn Poonen, Tate’s Thesis, MIT 18.786 lecture notes (2015); John Tate, Number theoretic background (1979), freely available paper; Pierre Deligne, Les constantes des équations fonctionnelles des fonctions L (1973), IAS archive.
13. Idèles in extensions and their cohomology
Read the lesson. Section 1 gives the local tensor decomposition from the primitive element and polynomial remainder lemmas of lesson 1. Subsequent sections prove norms, invariant class descent and the induced local calculation in the restricted product.
Free comparisons: Jürgen Neukirch, Class Field Theory — The Bonn Lectures, Online Edition 2.0 (May 2015), edited by Alexander Schmidt; Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
14. The Herbrand quotient of the idèle class group
Read the lesson. The local induced-module quotients and the proved S-unit representation give the idèle-class Herbrand quotient. The function-field compactness and lattice argument and the constant-field example are written separately with their full hypotheses.
Free comparisons: Jürgen Neukirch, Class Field Theory — The Bonn Lectures, Online Edition 2.0 (May 2015), edited by Alexander Schmidt; J. S. Milne, Class Field Theory, version 4.03; Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
15. The norm index bound and Hasse's norm theorem
Read the lesson. The Kummer norm-index bound includes Sylow descent. The function-field proof treats prime-to-characteristic radicals and characteristic-p additive duality separately. The cyclic Hasse norm theorem and conic principle retain their exact analytic and approximation prerequisites. The full Hasse–Minkowski argument is supplied after its ray-prime input in lesson 22, section 6A.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
16. The global reciprocity law
Read the lesson. The principal cancellation, local comparison and auxiliary procyclic construction are proved here. Number-field cyclotomic degree and function-field constant degree are constructed separately; the permitted profinite valuation image is retained.
Free comparisons: Jürgen Neukirch, Class Field Theory — The Bonn Lectures, Online Edition 2.0 (May 2015), edited by Alexander Schmidt; Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
17. Global existence and the idèlic class field correspondence
Read the lesson. The number-field S-unit existence argument and norm descent are retained. The characteristic-p radical and residue separation proof is written here, together with the complete function-field Weil extension and its topology.
Free comparisons: Jürgen Neukirch, Class Field Theory — The Bonn Lectures, Online Edition 2.0 (May 2015), edited by Alexander Schmidt; Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition; John Tate, Number theoretic background (1979), freely available paper.
18. Ray class fields, conductors and ideal reciprocity
Read the lesson. The local conductor criterion, optional real modulus, idèle-to-ideal ray identification, norm kernel, Takagi correspondence and decomposition law are proved with the stated fractional-ideal conventions.
Free comparisons: Jürgen Neukirch, Class Field Theory — The Bonn Lectures, Online Edition 2.0 (May 2015), edited by Alexander Schmidt; J. S. Milne, Class Field Theory, version 4.03.
19. Hilbert and ring class fields, and quadratic prime forms
Read the lesson. Sections 1–3 prove Hilbert fields, the augmentation-ideal transfer theorem, principalization and the ordinary/narrow genus distinction. Section 4 proves order arithmetic and reduction. Sections 6–8 verify the explicit quadratic prime-form fields by degrees, ramification and residue calculations; lesson 24 supplies the tower arguments.
Free comparisons: Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition; J. S. Milne, Class Field Theory, version 4.03.
20. Kronecker–Weber and the maximal abelian extension of the rationals
Read the lesson. The lesson proves both Kronecker–Weber routes, the rational Artin map, conductors and discriminants. Section 6 proves the cyclotomic values and the canonical topological component identification. The Bost–Connes state construction retains its separately identified programme prerequisite.
Free comparisons: J. S. Milne, Algebraic Number Theory; J. S. Milne, Class Field Theory, version 4.03; Alain Connes and Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives, author-posted draft.
21. Artin L-functions, conductors and discriminants
Read the lesson. The ramified induction identities, conductor integrality, conductor–discriminant identity and meromorphic functional equation are proved using the named internal character and Fourier lessons. The function-field analytic argument is written in section 6. The orthogonal deduction uses LG-GAL-06, Theorem 7.4, with its real-induction, Clifford-class and dihedral Fourier lemmas 7.5–7.8.
Free comparisons: Bjorn Poonen, Tate’s Thesis, MIT 18.786 lecture notes (2015); Wen-Wei Li, Yanqi Lake Lectures on Algebra: Part 1, author edition dated 2026-06-09; Pierre Deligne, Les constantes des équations fonctionnelles des fonctions L (1973), IAS archive; Pierre Deligne, Les constantes locales de l’équation fonctionnelle de la fonction L d’Artin d’une représentation orthogonale (1976), IAS archive.
22. The Chebotarev density theorem
Read the lesson. The lesson proves nonvanishing at 1, abelian character density and the cyclic fixed-field coset count, then Bauer and primes in ray classes. It proves Dirichlet density over both sorts of global field; no natural-density or effective assertion is substituted.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; Bjorn Poonen, Tate’s Thesis, MIT 18.786 lecture notes (2015); Kiran S. Kedlaya, Notes on class field theory, author-hosted HTML edition.
23. Power residue symbols and reciprocity laws
Read the lesson. The global Hilbert product and the tame formula give the general power-residue identity. The primary quadratic law keeps its real sign. The explicit cubic norm polynomial and quartic local two-generator pairing prove the primary cubic and quartic laws in the lesson.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03.
24. Brauer groups of local and global fields
Read the lesson. The local invariant, global Brauer exact sequence and Tate fundamental-class theorem are proved using the exact internal algebra and cochain prerequisites. The number-field Weil construction, complete filtered group-algebra inequality and arithmetic unit-cohomology relation bound are supplied here; a survey statement is not counted as their proof.
Free comparisons: J. S. Milne, Class Field Theory, version 4.03; John Tate, Number theoretic background (1979), freely available paper; Mikhail Ershov, Golod–Shafarevich groups: a survey, author preprint (2012).
Prerequisite proofs
The full statements, hypotheses and proof locators of the arithmetic, Fourier, character-theory, algebra and cochain prerequisites belong to their named programme lessons. An external source and a dependency record do not establish that a missing proof has been supplied.
In particular, lessons 12 and 21 use LG-GAL-06, Theorem 3.0 with sections 3A–3D and Theorem 7.4 with Lemmas 7.5–7.8. The deductions retain these explicit programme dependencies. The free Deligne papers are additional reading, not substitutes for these internal arguments.