This is the webpage for the course Topics in Algebraic Number Theory, taught at Leiden in the Spring of 2024 by Eugenia Rosu, Peter Stevenhagen and Victor de Vries. In this course, we will develop the foundations of the theory of local fields.
The main part of the course follows the notes written by Stevenhagen.
Lectures (Rosu/Stevenhagen) Snellius Fridays 11:00 - 12:45
Exercises (de Vries) Snellius/Gorlaeus Fridays 13:15 - 15:00
Registered students who are taking this course for credit are required to hand in weekly homework assignments, which count for 20% of the final grade. There is a take-home final exam that will be assessed in an oral examination (May 31st, June 21st 2024). The outcome constitutes 80% of the overall grade.
Homework policy:
Every Friday after the lecture, a short list of problems will be announced on this webpage, 2 of which are to be handed by email to vriesdevwh@vuw.leidenuniv.nl by 11am on Friday of the next lecture. There will be 12 homework sets, of which we count the 10 best scores. Presenting solutions is encouraged, and so is interaction with the presenters.
Lectures
Below is the schedule for the lectures.
Date Lecturer Subject
9 Feb 2024 Stevenhagen Motivation and overview
-
16 Feb 2024 Rosu Valued fields (continued). Complete fields
de Vries Exercises
Homework 1: Choose 2 exercises from: 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.11, 1.16, 1.18
23 Feb 2024 Stevenhagen Complete fields and local fields
de Vries Exercises
Homework 2: Choose 1 exercise from 1.10, 1.19, 1.20 and 1 exercise from 2.7-2.13
01 Mar 2024 Stevenhagen Hensel's Lemma
de Vries Exercises
Homework 3: Choose 2 exercises from
1.10, 1.19, 1.20, 2.8, 2.10-2-13 (last week's) and 2.18, 2.19.
Note: submit different exercises than you did last week.
08 Mar 2024 Rosu Extensions of valuations
de Vries Exercises
Homework 4: Choose 1 exercise from 2.14-2.16 and 1 exercise
from 2.19-2.23
15 Mar 2024 Stevenhagen Extensions of complete fields
de Vries Exercises
Homework 5: Choose 2 exercises from 3.3-3.11 and 3.13
22 Mar 2024 Stevenhagen Extensions of complete fields
de Vries Exercises
Homework 6: Choose 2 exercises from 3.14-3.19 and 4.1-4.4
5 Apr 2024 Rosu Galois theory of valued fields
de Vries Exercises
Homework 7: Choose 2 exercises from 4.3-4.12
12 Apr 2024 Rosu Galois theory of valued fields
de Vries Exercises
Homework 8: Choose 2 exercises from 5.2, 5.3, 5.9, 5.10, 4.7, 4.9, 4.10
26 Apr 2024 Rosu Galois theory of valued fields
de Vries Exercises
Homework 9: Choose 2 from exercises: 4.7,4.9, 4,10, 5.4, 5.7, 5.9, 5.11
3 May 2024 Rosu The Kronecker-Weber Theorem
de Vries Exercises
Homework 10: Choose 2 exercises from 5.6, 5.8, 5.13, 5.14, 5.15 and
exercise 5.5 from the old lecture notes https://websites.math.leidenuniv.nl/algebra/localfields.pdf
17 May 2024 Stevenhagen TBA
de Vries Exercises
Homework 11: Choose 2 exercises from 6.1-6.6
24 May 2024 Stevenhagen TBA
de Vries Exercises
Homework 12: Choose 1 exercise from: 8.3, 8.8, 8.9, 8.10, 8.12, 8.13, 8.14, 8.16, 8.18
Last 3 chapters of the notes by Stevenhagen
Details take-home exam:
Friday, 24 May 2024: take-home exam problems announced
Monday, 27 May 2024, 23:59h: take-home exam due. Please upload your solutions typed in LaTeX on Brightspace.
Wednesday, 29 May 2024: oral examination of take home exam
Retake exam: 21 June 2024