Math 125 Current Problems in Number Theory:
Galois Cohomology and Descent


Term: Winter 2022

Evariste Galois
Lecture (10636)
Inst : Prof. Asher Auel
asher * auel AT dartmouth * edu
Time : Tue Thu 10:10 am - 12:00 pm
X-hour Fri 3:30 - 4:20 pm
Loct : Kemeny 201
Office : Kemeny 339
Phone : None
Office
hours : 		TBA
 Course syllabus page.

Description of course: This course will explore some basic themes in field arithmetic and arithmetic geometry. Topics will include Galois cohomology and descent, cohomological dimension and Tsen-Lang theory, principal homogeneous spaces, quadratic forms and the Brauer group, Milnor K-theory, as well as the existence of rational points.

Expected background: Prior experience with algebra and Galois theory will be necessary. Some exposure to algebraic geometry will be useful.

Grading: Your grade will be based on class participation, collaborative group work, and a final project.

Group work, honestly: Working with other people on mathematics is highly encouraged and fun. You may work with anyone (e.g., other students in your section, in the course, not in the course, ...) on your homework problems. If done right, you'll learn the material better and more efficiently working in groups. The golden rule is:

Work with anyone on solving the problems,
but write up your final solutions by yourself.
Writing up the final draft is as important a process as figuring out the problems on scratch paper with your friends. If you work with people on a particular assignment, list your collaborators on the top of the first page. This makes the process fun, transparent, and honest. You will not learn (nor adhere to the Honor Principle) by copying solutions directly from others or from the internet.

COVID-19:

Attendance: You are expected to attend class in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community, please: do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home. Lecture notes will be made available to those who cannot attend class in person.

Masking: In accordance with current College policy, all members of the Dartmouth community are required to wear a mask when in our classroom, regardless of vaccination status. If you do not have an accommodation (see below) and refuse to comply with masking, I am obligated to ask you to leave the classroom and to report you to the Dean’s office for disciplinary action.

Accommodations: Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.