5230 Algebraic Number Theory

Algebraic Number Theory, Fall 2025

This course will cover the foundations of modern algebraic number theory.

Course information

Credits: 3
Prerequisites: MATH 5211.
Class meetings: TuTh 12:30-1:45am in Monteith 113

Instructor information

Professor: Rebecca Bellovin
E-mail: r dot m dot bellovin at the usual UConn domain
Office: Monteith 229
Office hours: TBA

Homework

Homework will generally be due every other week on Thursdays and cover material from the previous two weeks' lectures. You may either hand-write your solutions or type them (if you choose to type your solutions, I strongly urge you to learn LaTeX). It is only mandatory for undergraduates, but I strongly urge PhD students to do as many exercises as possible.

(Due 9/1l)
Ch. 1: 7, 8, 19–28
Ch. 2: 2, 4, 11, 12, 13
(Due 9/25)
Ch. 2: 8, 16, 19, 20, 24, 28, 30, 35, 41
(Due 10/14)
Ch. 1: 30-32. Prove that this ideal class group agrees with the definition in terms of fractional ideals.
Ch. 3: 8, 9, 11, 13, 14, 15, 16, 17
(Due 11/4)
Ch. 3: 33 – 37
Ch. 4: 5, 6, 7, 8, 12, 13
Extra problems on local fields
(Due 11/20)
Ch. 4: 18 – 23 (these make precise my comments from lecture about the filtration on decomposition groups)
Ch. 5: 8, 10, 26, 33, 35

Textbook

We will primarily follow Number Fields by Marcus. We will also refer to Neukirch's Algebraic Number Theory for some material (particularly on local fields).

Keith Conrad has extensive notes on Galois theory and algebraic number theory, which you may additionally find useful.

Assessment

Grades in this class will be based on homework. There will be no final exam.

Schedule

The following schedule is tentative and subject to change.

Week of	Topic(s)	Due date(s)
8/25	Introduction Number fields and rings of integers
9/1	Trace, norm, and discriminant
9/8	Integral bases for rings of integers	9/11: Homework 1
9/15	Commutative algebra
9/22	Dedekind domains Prime factorization	9/25: Homework 2
9/29	Inertia and ramification	10/2: No lecture
10/6	Valuations and completion
10/13	Local fields	10/14: Homework 3
10/20	Local fields (cont'd)
10/27	Ramification and discriminant
11/3	Class groups	11/4: Homework 4
11/10	Unit groups
11/17	Class number formula	11/20: Homework 5
11/24	Thanksgiving recess
12/1	TBD

University policies

Academic integrity

Please respect your work and the work of others. Cheating will be taken seriously. Examples of things that will be addressed include, but are not limited to, communicating with anyone not explicitly allowed during any quiz or exam, representing another person's work as your own (this includes copying or paraphrasing a solution from a friend, solution manual, tutor, or website), or bringing unauthorized materials to any quiz or exam. Consequences may include, but are not limited to, a score of zero on the assignment, quiz, or exam, or a grade of F in the course. To read UConn's full policy on Academic Integrity, visit \url{https://policy.uconn.edu/2023/07/11/academic-scholarly-and-professional-integrity-and-misconduct-aspim-policy-on/}.

Students with disabilities

The University of Connecticut is committed to protecting the rights of individuals with disabilities and assuring that the learning environment is accessible. If you anticipate or experience physical or academic barriers based on disability or pregnancy, please let me know immediately so that we can discuss options. Students who require accommodations should contact the Center for Students with Disabilities, Wilbur Cross Building Room 204, (860) 486-2020 or http://csd.uconn.edu.