Mathematisches Institut der Universität Bonn

V3A2/F4A1 Algebra II - winter term 2023/24

Dr. David Schwein

Contact:
Dr. Johannes Anschütz
Endenicher Allee 60 · Zimmer 4.027
Tel.: 0228-73-62216
E-mail: ja(add @math.uni-bonn.de)

Time and place

Monday, 16-18h, Kleiner Hörsaal
Thursday, 14-16h, Kleiner Hörsaal
First lecture: Thursday, 12.10.2023
End of lecture period: 2.2.2023

Content

A (nonarchimedean) local field is the Laurent series over a finite field or a finite extension of the p-adic numbers. Local fields play an important role in number theory and studying them will introduce us to several important ideas in mathematics. I plan to focus on

• Basic structure theory of local fields
• Profinite Galois theory
• Ramification theory and Newton polygons
• Galois cohomology of local fields
• Local class field theory

The prerequisites for this course are knowledge of basic algebra and Galois theory, basic topology, and integral extensions and discrete valuation rings, as covered (for instance) in

• V2A1 Einführung in die Algebra
• V2D1 Einführung in die Geometrie und Topologie
• V3A1/F4A1 Algebra I: Commutative Algebra.

It is not strictly necessary to have taken Algebra I.

Literature

The course will not follow any one text linearly, but will draw on the following sources.

• J. Neukirch and A. Schmidt, Class Field Theory. The Bonn Lectures. (2013; Zbl 1282.11001)
• J.-P. Serre, Local Fields. 4th corrected edition. (2004; Zbl 1095.11504)
• F. Q. Gouvêa, p-adic numbers. An introduction. (2020; Zbl 0786.11001)
• K. Conrad, Infinite Galois theory.

Exercise sheets

The weekly exercise sheets will be uploaded here.

• Exercise sheet 1 (11.10.2023)
• Exercise sheet 2 (19.10.2023)
• Exercise sheet 3 (26.10.2023)
• Exercise sheet 4 (2.11.2023)
• Exercise sheet 5 (9.11.2023)
• Exercise sheet 6 (corrected version) (16.11.2023)
• Exercise sheet 7 (24.11.2023)
• Exercise sheet 8 (30.11.2023)
• Exercise sheet 9 (7.12.2023)
• Exercise sheet 10 (14.12.2023)
• Exercise sheet 11 (11.01.2024)
• Exercise sheet 12 (18.01.2024)
• Exercise sheet 13 (25.01.2024)

Exercise sessions

In addition to the lecture there are weekly exercise sessions. Registration to the exercise sessions is via eCampus during the first week of the lectures. The exercise sessions themselves start in the second week of lectures. The registration via eCampus does NOT replace the registration to the course via Basis.

Group	Time and place	Tutor
Group 1	Monday, 8-10h, SR0.007	Ruth Wild
Group 2	Monday, 10-12h, SR0.007	Tanja Helme
Group 3	Tuesday, 8-10h, SR0.007	Giulio Ricci

The exercise sessions will take place in the buildings of the Mathematisches Zentrum, Endenicher Allee 60. Due to a lower number of participants than expected the fourth exercise group got cancelled.

Exam

First exam: Wednesday, 14.2.2024, 9:00-11:00h s.t., Großer Hörsaal. Revision: 15.2., 11:30-12:30h, N0.007
Second exam: Wednesday, 13.3.2024, 9:00-11:00h s.t., Großer Hörsaal. Revision: 14.3., 11-12h, N0.007
After the end of the lectures there will be a written exam. All necessary information will be provided via this webpage during the semester. For participation in the exam a registration via Basis is necessary as well a sucessful participation in the weekly exercise sessions (more than 50% of all points). General informations to exams can be found here for bachelor students and here for master students.

Last modified: September 2023, Johannes Anschütz