NMAG430
Summer semester 2021/22
Course description:
The course aims to introduce the ring of integers in the number fields and study its ideals and units. The main topics are the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory and introduction to p-adic numbers.
Outline:
Lecture 1 (14/02): Introduction - Integrality - Trace - Norm [See Neukirch - Ch. I, §2]
Lecture 2 (17/02): Discriminant [See Neukirch - Ch. I, §2]
Lecture 3 (21/02): Integral basis - [See Neukirch - Ch. I, §2]
Seminar 1 (24/02): Exercises - Solutions (Curated by Daniel Gil Muñoz)
Lecture 4 (28/02): Ideals - Dedekind domains - Unique prime factorisation - [See Neukirch - Ch. I, §3]
Lecture 5 (3/03): Chinese remainder theorem - Fractional ideals - Ideal class group - Lattices [See Neukirch - Ch. I, §3, §4]
Lecture 6 (7/03): Lattices - Minkowski's Lattice Point Theorem - [See Neukirch - Ch. I, §4]
Seminar 2 (10/03): Exercises - Solutions (Curated by Daniel Gil Muñoz)
Lecture 7 (14/03): Minkowski theory - [See Neukirch - Ch. I, §5]
Lecture 8 (17/03): The Class Number - [See Neukirch - Ch. I, §6]
Lecture 9 (21/03): Derichlet's Unit Theorem - [See Neukirch - Ch. I, §5, 7]
Seminar 3 (24/03): Exercises - Solutions (Curated by Daniel Gil Muñoz)
Lecture 10 (28/03): Extensions of Dedekind Domains - Ramification - [See Neukirch - Ch. I, §8]
Lecture 11 (31/03): Fundamental identity - [See Neukirch- Ch. I, §8]
Lecture 12 (5/04): Dedekind's factorization criterion (Notes - Video) - [See Neukirch - Ch. I, §8]
Seminar 4 (7/04): Exercises - Solutions (Curated by Daniel Gil Muñoz)
Lecture 13 (12/04): Ramification for Galois fields - Cyclotomic fields (Video) - [See Neukirch - Ch. I, §9, 10 and Milne Ch. 6]
Lecture 14 (19/04): Cyclotomic fields continued (Video) - [See Neukirch - Ch. I,10 and Milne Ch. 6]
Seminar 5 (21/04): Exercises - Solutions (Curated by Daniel Gil Muñoz)
Lecture 15 (26/04): Introduction to local fields (Video) - [Milne Ch. 7]
Lecture 16 (29/04): Equivalent absolute values (Video) - [Milne Ch. 7]
Lecute 17 (2/05): The weak approximation theory - [Milne Ch. 7]
Seminar 6 (5/05): Exercises - Solutions (Curated by Daniel Gil Muñoz)
Lecute 18 (12/05): Extensions of noncharimedean absolute values - [Milne Ch. 7]
Lecute 19 (16/05): Global fields - [Milne Ch. 8]
Seminar 7 (19/05): Exercises - Solutions (Curated by Daniel Gil Muñoz)
Recommended literature:
Algebraic Number Theory - J. Neukirch (Ch. I-II).
Algebraic Number Theory - J. S. Milne