Tags: #todo #todo/learning #subjects/NT #resources/references Refs: ?

Learning Algebraic Number Theory

References

Topics

Number fields and rings of integers;
Structure of the class and unit groups;
Splitting, ramification, and inertia of prime ideals under finite extensions;
Different and discriminant; basic properties of local fields.
[[Class field theory.md | Class%20field%20theory.html]]

Course

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.