2023/24

NMAG430

Summer semester 2023/24

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 (19/02):  Introduction - Embeddings - Trace - Norm [See Markus - Ch. 2]

Lecture 2 (22/02):  The Discriminant - Additive Structure of a Ring [See Markus - Ch. 2]

Lecture 3 (26/02):  Additive Structure of a Ring (continued) [See Markus - Ch. 2]

Seminar 1 (29/02): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 4 (04/03):  Prime Decomposition in Number Rings [See Markus - Ch. 3]

Seminar 2 (07/03): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 5 (11/03):  Ideals in Dedekind domains [See Markus - Ch. 3]

Seminar 3 (14/03): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 6 (18/03):  Splitting of Primes [See Markus - Ch. 3]

Lecture 7 (21/03):  Fundamental identity and ramified primes [See Markus - Ch. 3]

Lecture 8 (25/03):  Examples of prime decompositions [See Markus - Ch. 3]

Seminar 4 (28/03): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 9 (04/04):  Dedekind's criterion, Galois theory, and decomposition [See Markus - Ch. 3]

Lecture 10 (08/04):  Decomposition and inertia fields [See Markus - Ch. 4]

Seminar 5 (11/04): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 11 (15/04):  Quadratic reciprocity [See Markus - Ch. 4]

No class (18/04)

Lecture 12 (22/04): Frobenius automorphism [See Markus - Ch. 4]

Seminar 6 (25/04): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 13 (29/04): The ideal class group [See Markus - Ch. 5]

Lecture 14 (02/05): Minkowski's theorem [See Markus - Ch. 5]

Lecture 15 (06/05): The unit group [See Markus - Ch. 5]

Seminar 7 (09/05): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 16 (13/05): Local fields [See Milne - Ch. 7]

Seminar 8 (16/05): Exercises - Solutions (Curated by Stevan Gajović)

Lecture 17 (20/05): Ostrowski's Theorem, product formula, weak approximation [See Milne - Ch. 7]

Lecture 18 (23/05): Completion, Hensel's lemma [See Milne - Ch. 7]

Homework:

Homework 1 (due March 28)

Homework 2 (due April 25)

Homework 3 (due May 23)

Recommended literature:

Number Fields - D. A. Markus (Ch. 1-6). 

Algebraic Number Theory - J. S. Milne