# Math 373/573 Algebraic Number Theory

## Semester: Spring 2019

Lecture 01 (22060/23843)
 Inst : Prof. Asher Auel asher * auel AT yale * edu Time : Tue Thu 09:00 am-10:15 am Loct : LOM 214
 Office : LOM 210 Phone : (203) 432-4187 Officehours : Wed 2:30 - 3:30 pm
 Text : Algebraic Number Theory (v3.07) J. S. Milne Available at www.jmilne.org/math/
Course syllabus and homework schedule.

Description of course: The main object of study in Algebraic Number Theory are number fields (finite extensions of the field of rational numbers) and their rings of algebraic integers (those elements that are roots of monic polynomials with integer coefficients). Rings of algebraic integers have a theory of prime ideals that encode many interesting properties about the usual rational prime numbers, for example, the decomposition of primes in the Gaussian integers detects whether they can be written as a sum of two squares (like 5) or not (like 7). One of the most fundamental algebaic invariants of a number field is its ideal class group, which measures, among other things, the failure of unique factorization in the ring of integers. One of the fundamental theorems in Algebraic Number Theory is the finiteness of the ideal class group. Another is about the structure of units in the ring of integers. Both of these fundamental invariants are also mysteriously hidden in the Dedekind zeta function of a number field, analogous to the classical Riemann zeta function.

The topics covered will include: the ring of algebraic numbers; decomposition of ideals into products of prime ideals; the discriminant, different, and ramification theory; finiteness of the ideal class group; Dedekind unit theorem; the Dedekind zeta function; the p-adic numbers, ring of adeles, and group of ideles.

Expected background: Previous exposure to abstract algebra, field and Galois theory, and some complex analysis is required. For example, the contents of Math 350 Introduction to Abstract Algebra, Math 370 Fields and Galois Theory, and Math 310 Introduction to Complex Analysis, would suffice.

 Homework 50% Midterm exam 20% Final exam 30%
Grades: Your final grade will be based on weekly homework, two midterm exams, and a final exam (see formula). Equal overall emphasis is placed on exams and on homework assignments.
Group work, honestly: Working with other people on mathematics is highly encouraged and fun. You may work with anyone (e.g., other students in the course, not in the course, tutors, ...) 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 your homework problems,
but write up your final draft by yourself.
Writing up the final draft is as important a process as figuring out the problems on scratch paper with your friends, see the guidelines below. Mathematical writing is very idiosyncratic - we will be able to tell if papers have been copied - just don't do it! You will not learn by copying solutions from others or from the internet! Also, if you work with people on a particular assignment, you must list your collaborators on the top of the first page. This makes the process fun, transparent, and honest.

# Policies

(or otherwise the small print)

Homework: Each assignment will be posted on the syllabus.

If you know in advance that you will be unable to submit your homework by the deadline, let me know.

Do not copy homework solutions from internet resources!

Your lowest homework score above 50% from the semester will be dropped.

Exams The midterm exam will take place in-class TBA. The final exam with be a take home exam.

Make-up exams will only be allowed with a dean's excuse.

Homework guidelines: Generally, a homework problem in any math course will consist of two parts: the creative part and the write-up.

• The creative part: This is when you "solve" the problem. You stare at it, poke at it, and work on it until you understand what's being asked, and then try different ideas until you find something that works. This part is fun to do with your friends; you can do it on the back of a napkin. If you're having trouble, even in understanding what the problem's asking, use the resources available to you: my office hours, teaching assistants' office hours, weekly tutoring sessions, etc. Ask for help as early as you can! This part should all be done on "scratch paper."

• The write-up: Now that everything about the problem is clear in your mind, you go off by yourself and write up a coherent, succinct, and nicely written solution on clean sheets of paper. Consider this your final draft, just as in any other course. This part you should definitely NOT do with your friends.