Spring 2023
Math 129: Number fields
E-mail: tayou@math.harvard.edu ,
Office: Science Center 238.
Course assistant:
E-mail: ,
Office: Science Center .
Schedule:
- Meeting times: Wednesday-Friday 09:00 AM-10:15 AM.
- Room: Science Ctr 221.
- First meeting: Wednesday, January 25.
- Office hours: Wednesday 5-6PM or by appointment, in Science Center 238.
Syllabus:
This class is an advanced undergraduate course and it is an introduction to algebraic number theory. Number fields are fundamental objects of study in number theory and algebraic geometry. They are ubiquitous in several areas of mathematics.
We will first start by studying unique factorization of ideals in number fields, we will define the Picard group of the ring of integers of a number field and prove that it is a finite group. Our next object of study will be the structure of the units group, the structure of the Galois group, the Frobenius elements, and ramification theory. The final topic will be an introduction to analytic number theory: after introducing the Dedekind Zeta function, we will prove the class number formula. If time permits, we will introduce the adeles and the ideles.
Recommended books:
The main references will be:
- Daniel A. Marcus, "Number fields".
- Pierre Samuel, "Algebraic Theory of Numbers".
For more advanced reading and exercises, I recommend the following book:
- Jürgen Neukirch, "Algebraic number theory".
- Gerald J.Janusz, "Algebraic Number fields".
Prerequisites:
Math 122-123 (Algebra I-II) or equivalent.
Grading:
Homework will be assigned weekly. The solutions can either be scanned or typed and uploaded on Canvas. Homework will count for at least 70% of the final grade. Late homework can only be accepted under special circumstances. Collaborative work on homework is accepted but you must write your own solution as well as the names of the collaborators. There will be a midterm and a final exam, which will count for 30% of the grade.