Prerequisite proofs and availability
All twenty-four lessons and ninety-six solved exercises are available. Some proofs use results from earlier programme courses. This page identifies the exact results and their hypotheses; it does not contain those prerequisite proofs. External sources remain useful reading and do not supply an internal proof by citation alone. A written prerequisite whose reader is not yet published remains an explicit dependency for online readers. The course is not claimed to have a fully closed prerequisite chain.
Lesson 1
Lesson 6
Lesson 7
NT-LOC-04, NT-LOC-07, NT-LOC-02.
Lesson 8
Lesson 9
Lesson 10
NT-LOC-07, NT-LOC-09, NT-LOC-10.
Lesson 11
Lesson 12
NT-ADL-07, NT-ADL-08, LG-GAL-06.
Lesson 13
Lesson 14
NT-LOC-01, NT-ADL-03, NT-ANT-09.
Lesson 15
Lesson 19
Lesson 20
NT-ANT-12, NT-ADL-06, QSM-L04, QSM-L03.
Finite character extension and separation are proved before section 4 of lesson 3. Proposition 20.5 proves the least-conductor assertion directly by Chinese remainders and the local unit filtration.
Lesson 21
NT-ADL-10, RT-FIN-02, RT-FIN-06, RT-FIN-11, LG-GAL-06.
Lesson 22
Lesson 24
NOE-HYP-04, NOE-HYP-05, ag-etale-cohomology-08.
NT-ANT-12: Cyclotomic fields
Exact result: Theorems 12.1 and 12.3.
Cyclotomic irreducibility, degree and Galois group; inertia and arithmetic Frobenius for every prime, including the factor 2 of a modulus congruent to 2 modulo 4. The existing supporting reading supplies Theorem 12.1 only.
The full result-level proof is supplied in the existing NT-ANT lesson cyclotomic-fields. Its proof is not given in these courses.
Read the already published full proof of Theorem 12.1. That excerpt does not include Theorem 12.3.
Used in NT-CFT-01, NT-CFT-20.
NT-LOC-04: Extensions of complete valued fields
Exact result: Theorem 1.2, Proposition 2.1, Theorem 3.1.
Finite extensions of complete valued fields have a unique extended absolute value and are complete; coordinate norms are equivalent. For complete discrete valuation fields the residue-digit integral basis gives degree ef, including inseparable extensions and imperfect residues.
The full result-level proof is supplied in the existing NT-LOC lesson NT-LOC-04. Its proof is not given in these courses.
Used in NT-CFT-06, NT-CFT-07, NT-CFT-08.
NT-LOC-07: Unramified and totally ramified extensions
Exact result: Theorem 2.1, Corollary 3.1, Proposition 4.1, Theorem 5.1.
Unramified residue lifting uses finite separable residue extensions, without a perfectness assumption. The cyclic tower and surjective unit norm use finite residue fields. Eisenstein extensions are totally ramified over a complete discrete valuation field.
The full result-level proof is supplied in the existing NT-LOC lesson NT-LOC-07. Its proof is not given in these courses.
Used in NT-CFT-06, NT-CFT-07, NT-CFT-08, NT-CFT-09, NT-CFT-10, NT-CFT-11, NT-CFT-13.
NT-LOC-02: Completions, the p-adic numbers and complete discretely valued fields
Exact result: Theorem 2.1 and Proposition 2.2.
Completion and extension of isometries; integer density and residue approximation in the p-adic integers. No claim that an arbitrary infinite algebraic extension is complete.
The full result-level proof is supplied in the existing NT-LOC lesson NT-LOC-02. Its proof is not given in these courses.
Used in NT-CFT-07.
NT-LOC-09: Ramification groups and the different of a local extension
Exact result: Proposition 1.1, equation (2.2), Theorem 3.1.
Lower ramification groups and Hilbert different formula for finite Galois local extensions; normalized integer valuation and the inertia term are retained.
The full result-level proof is supplied in the existing NT-LOC lesson NT-LOC-09. Its proof is not given in these courses.
Used in NT-CFT-10.
NT-LOC-10: Herbrand's function and the upper numbering
Exact result: Proposition 1.1 and Theorem 3.1.
Herbrand quotient compatibility and upper numbering for finite Galois local extensions. The proof uses coset motion and tower composition, without assuming Hasse–Arf.
The full result-level proof is supplied in the existing NT-LOC lesson NT-LOC-10. Its proof is not given in these courses.
Used in NT-CFT-10.
NT-LOC-01: Absolute values, valuations and Ostrowski's theorem
Exact result: Lemma 5.1 and Theorem 5.2.
Weak approximation for finitely many inequivalent nontrivial absolute values on an arbitrary field; this covers the function-field use in lesson 14.
The full result-level proof is supplied in the existing NT-LOC lesson NT-LOC-01. Its proof is not given in these courses.
Used in NT-CFT-14.
NT-ADL-07: Tate's local theory at the finite places
Exact result: Proposition 7.1, Theorems 7.2–7.3, Proposition 7.4.
Finite-place rank-one Tate integrals and their functional equation, primitive Gauss sums and changes of measures and additive character. Its number-field local presentation is extended to equal characteristic in NT-CFT-12, section 4.
The full result-level proof is supplied in the existing NT-ADL lesson NT-ADL-07. Its proof is not given in these courses.
Used in NT-CFT-12.
NT-ADL-08: Tate's local theory at the infinite places
Exact result: Theorem 8.1 and Propositions 8.2–8.3.
Rank-one real and complex Schwartz integrals. Negative trace character and self-dual measure give phases (-i)^epsilon and (-i)^|n|; the complex angular parameter uses its absolute value.
The full result-level proof is supplied in the existing NT-ADL lesson NT-ADL-08. Its proof is not given in these courses.
Used in NT-CFT-12.
NT-ADL-03: Idèles and the idèle class group
Exact result: Propositions 3.1–3.2, Theorem 3.3, Corollary 3.4.
Restricted-product topology, principal idèles, number-field norm-one compactness and units. NT-CFT-13 uses the topology rather than importing a number-field-only cohomology theorem; NT-CFT supplies its own function-field arguments.
The full result-level proof is supplied in the existing NT-ADL lesson NT-ADL-03. Its proof is not given in these courses.
Used in NT-CFT-13, NT-CFT-14.
NT-ANT-09: Dirichlet's unit theorem
Exact result: Theorems 9.2 and 9.4.
Dirichlet units and S-units for number fields with S finite and containing every infinite place. The approximation-lattice proof and valuation exact sequence are supplied.
The full result-level proof is supplied in the existing NT-ANT lesson dirichlets-unit-theorem. Its proof is not given in these courses.
Used in NT-CFT-14.
NT-ADL-10: Hecke L-functions and the Dedekind zeta function
Exact result: Theorems 10.1–10.2 and Corollary 10.4; preceding Theorem 9.2.
Completed Hecke L-functions of all unitary number-field Hecke characters; only norm twists allow the two shifted simple poles. Intrinsic conductor is |d_K| Norm(f), with the local phases above. In particular nontrivial finite-order characters are holomorphic at 1 and the Dedekind zeta residue is positive. The global proof is in NT-ADL-09 and the local computations in NT-ADL-07/08.
The full result-level proof is supplied in the existing NT-ADL lesson NT-ADL-10. Its proof is not given in these courses.
Used in NT-CFT-15, NT-CFT-21, NT-CFT-22.
NT-ANT-11: Orders in number fields and their Picard groups
Exact result: Theorems 11.3–11.4 and Corollary 11.5.
Extension–contraction for ideals prime to the conductor, Picard-group exact sequence and quadratic order class-number formula. NT-CFT-19, section 4 also proves the calculations needed for its examples.
The full result-level proof is supplied in the existing NT-ANT lesson orders-in-number-fields-and-their-picard-groups. Its proof is not given in these courses.
Used in NT-CFT-19.
NT-ANT-10: Quadratic fields: ideal classes and binary quadratic forms
Exact result: Theorems 10.1–10.2 and Proposition 10.3.
Oriented quadratic ideal–form correspondence uses narrow classes; positive definite reduction includes the boundary convention. The real narrow-to-ordinary kernel depends on whether a fundamental unit has norm -1.
The full result-level proof is supplied in the existing NT-ANT lesson quadratic-fields-ideal-classes-and-binary-quadratic-forms. Its proof is not given in these courses.
Used in NT-CFT-19.
NT-ADL-06: Quasi-characters and Hecke characters
Exact result: Theorem 6.3, Propositions 6.2 and 6.4.
Ideal/idèle character correspondence, finite conductors and finite-order ray characters. The finite unit character is the inverse of the corresponding ideal character, matching arithmetic reciprocity.
The full result-level proof is supplied in the existing NT-ADL lesson NT-ADL-06. Its proof is not given in these courses.
Used in NT-CFT-20.
NT-ANT-16: The Dedekind zeta function and the analytic class number formula
Exact result: Theorem 16.2.
Positive simple Dedekind zeta pole at 1 from the earlier ideal-counting estimate, by a holomorphic error integral on Re(s)>1-1/[K:Q]. It does not require Chebotarev or full-plane continuation.
The full result-level proof is supplied in the existing NT-ANT lesson the-dedekind-zeta-function-and-the-analytic-class-number-formula. Its proof is not given in these courses.
Used in NT-CFT-22.
RT-FIN-02: Characters and the orthogonality relations
Exact result: Theorem 3.2.
For a finite group over the complex numbers, each irreducible occurs in the regular representation with multiplicity its dimension. The proof uses the regular trace and the preceding character orthogonality.
Read the supplied programme proof. The exact published source and proof locators are recorded in the RT-FIN proof crosswalk.
Used in NT-CFT-21.
RT-FIN-06: Induced representations and Frobenius reciprocity
Exact result: Sections 1–4.
Finite-group complex induction in tensor and equivariant-function models, Frobenius reciprocity, induced characters and induction in stages. The left tensor action and right translation on equivariant functions match the convention used in NT-CFT-21.
Read the supplied programme proof. The exact published source and proof locators are recorded in the RT-FIN proof crosswalk.
Used in NT-CFT-21.
RT-FIN-11: Brauer's induction theorem
Exact result: Theorem 5.1, with its proof in sections 1–5.
Every finite-group complex virtual character is an INTEGER combination of characters induced from one-dimensional characters of subgroups. The local elementary-group ideal argument and monomiality supply integral induction, not just rational Artin induction. Its earlier representation-theory prerequisites remain with RT-FIN.
Read the supplied programme proof. The exact published source and proof locators are recorded in the RT-FIN proof crosswalk.
Used in NT-CFT-21.
NOE-HYP-04: Central simple algebras and the Brauer group
Exact result: Theorems 4.2–4.3 and 5.1.
Finite splitting criterion, separable maximal subfields and the Brauer group for arbitrary fields. The separable-subfield proof treats characteristic zero, finite fields and infinite fields of positive characteristic.
Published complete programme proofs: Theorem 4.2; Theorem 4.3; Theorem 5.1. These Noether-course lessons are available as supporting readings in the ℓ-adic cohomology course. Matching editable source.
Used in NT-CFT-24.
NOE-HYP-05: Crossed products and factor systems
Exact result: Theorems 3.2 and 4.1, Proposition 6.2.
Finite Galois crossed products identify H^2(G,L×) with Br(L/K), compatibly with inflation. The cyclic parameter uses the specified generator; the right-module descent multiplier is the inverse cocycle, so the Brauer sign matches NT-CFT-24.
Published complete programme proofs: Theorem 3.2; Theorem 4.1; Proposition 6.2. These Noether-course lessons are available as supporting readings in the ℓ-adic cohomology course. Matching editable source.
Used in NT-CFT-24.
ag-etale-cohomology-08: Galois cohomology and the étale cohomology of a field
Exact result: Sections 2–3 and Lemmas 5.1–5.2.
Continuous cochains for discrete profinite modules, finite-quotient passage and Shapiro by exact coinduction for closed subgroups. For the finite groups in NT-CFT-24 these are ordinary cochains; its additional Tate constructions are proved there.
Read the supplied programme proof.
Used in NT-CFT-24.
LG-GAL-06: Local L-factors and epsilon-factors of Weil group representations
Exact result: Theorem 3.0, sections 3A–3D, Theorem 3.2; Theorem 7.4 and Lemmas 7.5–7.8.
Theorem 3.0 and sections 3A–3D supply local epsilon existence for finite-dimensional smooth complex Weil representations, including arbitrary unramified twists, trace-induced additive characters, virtual dimension-zero induction, measure scaling and rank-one geometric reciprocity. Theorem 3.2 supplies uniqueness. Theorem 7.4, proved using Lemmas 7.5–7.8, treats virtual finite-image real representations of dimension zero and determinant one, including characteristic two and infinite places. NT-CFT rank-one and global analytic proofs do not use these higher-dimensional inputs; the orthogonal global deduction does. Supplied arguments are distinguished from independent review and recursive proof closure.
Read the existence and uniqueness arguments and the orthogonal theorem and its supporting lemmas. The exact source and reader revision is recorded in the machine-readable prerequisite record. Independent review and recursive prerequisite closure are not claimed.
Used in NT-CFT-12, NT-CFT-21.
QSM-L04: Phase transition in the Bost–Connes system
Exact result: Sections 3, 4, 6 and 18.
Optional Bost–Connes connection: Gibbs states for beta>1 and their cooled limit beta→infinity have arithmetic values e(vr). The number-theoretic equivariance proved in NT-CFT-20 does not prove state positivity, the KMS classification or their operator-algebra prerequisites.
Optional connection: the lesson named in the heading contains the indicated discussion and proofs, with its separate operator-algebra prerequisites. Its programme reader is not yet available in this collection.
Used in NT-CFT-20.
QSM-L03: The Bost–Connes Hecke algebra
Exact result: Section 6.
Optional operator-algebra symmetry uses the compatible unit as cyclotomic exponent. Arithmetic idèle coordinates require the inverse. Its operator-algebra proofs and dependencies stay in QSM-L03.
Optional connection: the lesson named in the heading contains the indicated discussion and proofs, with its separate operator-algebra prerequisites. Its programme reader is not yet available in this collection.
Used in NT-CFT-20.
Foundation proofs in this course
Lesson 1, section 0, proves the finite automorphism lemma, finite Galois correspondence, primitive element theorem and the integer and polynomial Chinese remainder lemmas. Section 3 proves algebraic embedding extension. Lesson 2, section 1, proves its exact cyclic integral resolution. The finite character lemma before section 4 of lesson 3 proves character extension, cardinality and separation.
The remaining linear algebra, set theory and topology foundations belong to the earlier programme. This page does not certify their recursive prerequisite chain. Each externally sourced prerequisite also needs an eligible freely accessible source; a written programme file alone is not evidence of that source qualification.
The order of the concluding proofs
The full Hasse–Minkowski theorem is Theorem22.7, after the ray-prime theorem it uses. The genus construction is in lesson19; the complete infinite-tower conclusion is Theorem24.9, after the filtered Golod–Shafarevich and terminal unit-cohomology bounds. Both original statements and their full scope are retained in these proved homes.