Tags: #todo #subjects/ANT Refs: ?
Algebraic Number Theory
References
Texts
- Neukirch
Good exposition. First 2.5 chapters contains most of the core material. Supplement with problems from Marcus
- Marcus, Number Fields
Good source of computational problems
- Serre, Local Fields
- Lots of advice: http://www.mathcs.emory.edu/~dzb/advice.html
Notes
- See Definitions
- this MO question
- File: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
- Milne: https://jmilne.org/math/CourseNotes/ANTc.pdf
- https://kskedlaya.org/cft/preface-1.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;
-
Main statements of 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 Kronecker-Webber theorem
- Hilbert 90
- Kummer theory
- Krasner's lemma
- Weak approximation
- Strong approximation
- 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 #subjects/ANT which seems to motivate a lot of results in #subjects/CA
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.