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- Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - 2021 Class Field Theory Notes - Unsorted/Galois theory - analytic class number formula

MOC Algebraic Number Theory

References

Texts

• Marcus, Number Fields

  Good source of computational problems

• Local Fields, J.-P. Serre.

• Class Field Theory, J.S. Milne.\

• Algebraic number theory, J.W.S. Cassels and A. Frohlich. (errata).\

• Multiplicative Number Theory, H. Davenport.\

• Algebraic Number Theory, J.S. Milne.\

• Algebraic Number Theory, S. Lang.\

• Introduction to Modern Theory,Yu. I. Manin and A. A. Panchishkin.\

• Algebraic Number Theory, J. Neukirch.

Good exposition. First 2.5 chapters contains most of the core material. Supplement with problems from Marcus - A Course in Arithmetic, J.-P. Serre.\

• Szamuely, Galois Groups and Fundamental Groups

Notes

• See Definitions
• this MO question
• attachments/ant.pdf
• https://www.math.ucla.edu/~sharifi/algnum.pdf
• https://wstein.org/books/ant/ant.pdf
• https://math.berkeley.edu/~apaulin/NumberTheory.pdf
• https://kskedlaya.org/cft/preface-1.html
  • https://math.ucsd.edu/~kkedlaya/math204b/
• Lots of advice: http://www.mathcs.emory.edu/~dzb/advice.html

Topics

• What is Gauss’ lemma?
• What is the Galois correspondence?
• Number fields and ring of integers;
• Splitting, ramification, inertia of prime ideals under finite extensions
  • efd theorem
• Main statements of Unsorted/class field theory:
  • Galois cohomology
• What is the Hasse–Minkowski theorem?
• What is Artin symbol?
• What is the product formula for global fields?
• The ideal class group is finite.
• The unit group is finitely generated.
• Dirichlet’s unit theorem
• Minkowski’s Lattice Point Theorem
• The Minkowski bound
• The Unsorted/Kronecker-Weber theorem
• Hilbert 90
• Kummer theory
• Krasner’s lemma
• Weak approximation
• Strong approximation
• Unsorted/Chebotarev density

Analytic NT

• Dirichlet’s theorem on primes in APs
• The Riemann-Zeta function
• Poisson summation
• The Fourier transform
• Jacobi theta
• The Gamma function
• The analytic class number formula

Courses

MIT OCW in ANT

A useful course in #NT/algebraic which seems to motivate a lot of results in #CA

• 1 Absolute values and discrete valuations (PDF)

• 2 Localization and Dedekind domains (PDF)

• 3 Properties of Dedekind domains, ideal class groups, factorization of ideals (PDF)

• 4 Étale algebras, norm and trace (PDF)

• 5 Dedekind extensions (PDF)

• 6 Ideal norms and the Dedekind-Kummer theorem (PDF)

• 7 Galois extensions, Frobenius elements, and the Artin map (PDF)

• 8 Complete fields and valuation rings (PDF)

• 9 Local fields and Hensel’s lemmas (PDF)

• 10 Extensions of complete DVRs (PDF)

• 11 Totally ramified extensions and Krasner’s lemma (PDF)

• 12 The different and the discriminant (PDF)

• 13 Global fields and the product formula (PDF)

• 14 The geometry of numbers (PDF)

• 15 Dirichlet’s unit theorem (PDF)

• 16 Riemann’s zeta function and the prime number theorem (PDF)

• 17 The functional equation (PDF)

• 18 Dirichlet L-functions and primes in arithmetic progressions (PDF)

• 19 The analytic class number formula (PDF)

• 20 The Kronecker-Weber theorem (PDF)

• 21 Class field theory: ray class groups and ray class fields (PDF)

• 22 The main theorems of global class field theory (PDF)

• 23 Tate cohomology (PDF)

• 24 Artin reciprocity in the unramified case (PDF)

• 25 The ring of adeles, strong approximation (PDF)

• 26 The idele group, profinite groups, infinite Galois theory (PDF)

• 27 Local class field theory (PDF)

• 28 Global class field theory and the Chebotarev density theorem (PDF)

Kedlaya, Class Field Theory

Topics by date (with videos, references, notes, and boards): See also https://mediaspace.ucsd.edu/playlist/dedicated/1_mudvfogg/ for all of the videos at once.

• Oct 2 (F): Overview of the course (https://math.ucsd.edu/~kkedlaya/math204a/algebraic_numbers.pdf.
• Oct 5 (M): Gaussian and Eisenstein integers (https://brilliant.org/wiki/gaussian-integers/.
• Oct 7 (W): Eisenstein and other quadratic integers (https://miro.com/app/board/o9J_kkyj6FU=/)).
• Oct 9 (F): Rings of integers in number fields (https://brilliant.org/wiki/algebraic-number-theory/ > ring-of-integers.
  This lecture was not fully recorded due to a technical issue.
• Oct 12 (M): Unique factorization of ideals (https://miro.com/app/board/o9J_kkyj76U=/)). References: Neukirch I.2, I.3.
• Oct 14 (W): Discriminant of a basis, proof of unique factorization, fractional ideals (https://miro.com/app/board/o9J_kkyhUik=/)). References: Neukirch I.2, I.3.
• Oct 16 (F): The lattice of a number field (https://miro.com/app/board/o9J_kkyhUtM=/)). References: Neukirch I.5.
• Oct 19 (M): Minkowski’s theorem (https://miro.com/app/board/o9J_kkyipr0=/)). References: Neukirch I.4, I.5, I.6.
• Oct 21 (W): The class number; the multiplicative lattice of a number field (https://miro.com/app/board/o9J_kkyg1wg=/)). References: Neukirch I.5, I.6, I.7.
• Oct 23 (F): The multiplicative lattice and the units theorem (https://miro.com/app/board/o9J_kkyg1z0=/)). References: Neukirch I.6, I.7.
• Oct 26 (M): Computational tools for algebraic number theory (https://miro.com/app/board/o9J_kkyvON0=/)).
• Oct 28 (W): Extensions of Dedekind domains (https://miro.com/app/board/o9J_kkyvOPA=/)). References: Neukirch I.8.
• Oct 30 (F): continuation (https://miro.com/app/board/o9J_kkyvOJM=/)).
• Nov 2 (M): Cyclotomic fields (https://miro.com/app/board/o9J_kkyvOK8=/)). References: Neukirch I.10, Marcus chapter 2.
• Nov 4 (W): Galois groups, ramification, and splitting (https://miro.com/app/board/o9J_kkyvOL8=/)). References: Neukirch I.9.
• Nov 6 (F): continuation (https://miro.com/app/board/o9J_kkysR9o=/)).
• Nov 9 (M): Localization (https://miro.com/app/board/o9J_kkysO_k=/)). References: Neukirch I.11.
• No lecture on Wednesday, November 11.
• Nov 13 (F): continuation (https://miro.com/app/board/o9J_kkysOGs=/)).
• Nov 16 (M): Different and discriminant (https://miro.com/app/board/o9J_kkysOAQ=/)). References: Neukirch III.2.
• Nov 18 (W): continuation (https://miro.com/app/board/o9J_kkysOB8=/)).
• Nov 20 (F): Structure of ramification groups (https://miro.com/app/board/o9J_kkysOD4=/)). References: Neukirch II.10.
• Nov 23 (M): p-adic numbers (https://miro.com/app/board/o9J_kkysOMg=/)). References: Neukirch II.1.
• Nov 25 (W): p-adic absolute value (https://miro.com/app/board/o9J_kkysOO4=/)). References: Neukirch II.2, II.4.
• No lecture on Friday, November 27.
• Nov 30 (M): Valuations (https://miro.com/app/board/o9J_kkysOI8=/)). References: Neukirch II.3.
• Dec 2 (W): Extensions of valuations (https://math.ucsd.edu/~kkedlaya/math204a/extension_valuations.pdf.
• Dec 4 (F): Hensel’s lemma (https://miro.com/app/board/o9J_kkysOL4=/)). References: Neukirch II.4.
• Dec 7 (M): Newton polygons (https://miro.com/app/board/o9J_kkysOUQ=/)). References: Neukirch II.6.
• Dec 9 (W): The Kronecker-Weber theorem: preview of Math 204B (https://kskedlaya.org/cft/, chapter 1.
• Dec 11 (F): The local Kronecker-Weber theorem (https://kskedlaya.org/cft/, chapter 1.