Topics in Algebraic Number Theory

Leiden University, Winter 2025

This is the course webpage for Topics in Algebraic Number Theory (4373TOANT), taught at Leiden in Winter 2025 by Peter Stevenhagen (psh), Jonathan Love (j.r.love), and Alexander Molyakov (a.molyakov) (@math.leidenuniv.nl).

In this course, we will develop the theory of local fields. The main part of the course will follow the course notes written by Stevenhagen. For the last two lectures on adeles and class field theory, see for instance these notes by Lenstra and Kedlaya.

Update 28 May: Oral exam today is in Peter Stevenhagen's office: BW.1.47.

Course structure

Lectures: Fridays, 11:00 - 12:45, Gorlaeus Building (see schedule for location)
Exercises: Fridays, 13:15 - 15:00, Gorlaeus Building (see schedule for location)

Registered students who are taking the course for credit will be evaluated based on weekly homework assignments and a final exam.

Homework (20% of final grade): Every Friday after the lecture, a short list of problems will be announced on this webpage. You may choose any one or two problems (the number may change each week) from this list to submit before the start of the following lecture. Your two lowest homework scores will be dropped. In the exercise sessions you will be encouraged to present solutions and to interact with the presenters.

Send homework solutions to topicsinant2024@gmail.com.

Final Exam (80% of final grade): There will be a take-home final exam that will be assessed in an oral examination (Wednesday, 28 May 2025). Problems will be made available a few days beforehand. More details coming soon.

Schedule

Room locations: L for morning lecture, E for afternoon exercise session. Room locations may also be confirmed on MyTimetable.

Sometimes the same problem will be an option for multiple weeks' homework. You should ensure that you submit different problems every week.
Date/Rooms Lecturer Topics
7 Feb 2025
L: EM.1.09		 Stevenhagen Week 1: Motivation and overview
Homework: Start reading chapter 1 of the lecture notes as a preview for next lecture. Submit any one problem from Exercise 1.1~1.12 by 14 Feb.
14 Feb 2025
L: BW.0.29
E: BW.0.06		 Love
 Week 2: Valued fields
Homework: Submit any one problem from Exercise 1.8~1.16 by 21 Feb.
21 Feb 2025
L: BW.0.29
E: BW.0.06		 Love Week 3: Valued fields (continued)
Homework: Submit any one problem from Exercise 1.17~1.20 by 28 Feb.
28 Feb 2025
L: BW.0.29
E: BW.0.06		 Love Week 4: Completions
Submit any two problems from Exercise 2.1~2.11 by 7 Mar.
7 Mar 2025
L: CM.1.26
E: BW.0.19		 Stevenhagen Week 5: Local Fields and Hensel's Lemma
Submit any one problem from Exercise 2.12~2.23 by 14 Mar.
14 Mar 2025
L: BM.1.26
E: BW.0.06		 Stevenhagen Week 6: Extending Valuations
Submit any two problems from the following set by 21 Mar: Chapter 2 exercise 19~23, Chapter 3 exercise 3, 5, 7, 9, 11, 13, 15~19.
21 Mar 2025
L: BW.0.29
E: BW.0.06		 Stevenhagen Week 7: e and f
Submit any two problems from Chapter 3 Exercise 3, 5, 7, 9, 11, 13, 15~19 by 4 Apr.
28 Mar 2025 No course meeting
4 Apr 2025
L: BW.0.29
E: BW.0.06		 Stevenhagen Week 8: Extensions of complete fields
Submit any one problem from Exercise 4.1~4.12 by 11 Apr.
11 Apr 2025
L: DM.1.09
E: BW.0.05		 Love Week 9: Different and discriminant; decomposition, inertia, and ramification groups
Submit any one problem from Exercise 4.1~4.13, 4.15, 5.1, 5.2, 5.4 by 25 Apr.
18 Apr 2025 No course meeting
25 Apr 2025
L: BW.0.19
E: BW.0.06		 Stevenhagen Week 10: Galois theory and Frobenius elements
Submit any one problem from Exercise 4.13, 4.15, 5.1, 5.2, 5.4, 5.7, 5.8~5.11 by 2 May.
2 May 2025
L: BW.0.31
E: BW.0.06		 Stevenhagen Week 11: Kronecker-Weber
Submit any one problem from Exercise 5.12, 5.14, 5.15, 6.2~6.5 by 9 May.
9 May 2025
L: BW.0.31
E: BW.0.06		 Love Week 12: Adeles and ideles
Taken from Kedlaya's Notes. Submit any one problem from Section 6.1 Exercises 2, 4, 5, 6, Section 6.2 Exercises 1 or 2, or Section 6.3 Exercise 1.
12 May 2025 (Monday)
L: BW.0.30		 Love Week 13: Class Field Theory
16 May 2025
E: BW.0.06		 Week 13 Exercise Session
28 May 2025
(Wednesday)
BW.1.47		 Oral assessment of take-home exam
Last Updated February 10 2025