Math 7315 (Algebraic Number Theory), Fall 2024
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
MWR 4:35pm-5:40pm 049 Snell Library |
MW 3:00pm-4:00pm Lake 571 |
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| This course will provide a graduate-level introduction to algebraic number theory. Here is the course syllabus. | |||||
| Course Motivation: Classical elementary number theory consists of studying the properties of arithmetic in Q and its associated ring of integers Z (e.g., prime factorization, Fermat's little theorem, quadratic reciprocity, the prime number theorem, etc.). Algebraic number theory extends the study of these topics to the larger class of number fields, which are the finite algebraic extensions K of Q, using the tools of modern algebra. A broad theme is to study the presence (and failure) of unique factorization, with a particular highlight being the study of prime ideals and their splitting in extensions using Galois theory. Analytic tools such as the zeta function and more general L-functions also enter naturally into the discussion of questions about the distribution of primes. By combining all of these approaches, our aim is to extend many classical results of number theory into more general settings. | |||||
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Course Topics: We begin by motivating the general study of unique factorization in number fields using some classical problems in number theory (e.g., Fermat's equation). We then establish a number of basic properties of number fields and their associated rings of integers using algebra: we review Dedekind domains, then discuss traces, norms, discriminants, unique factorization of ideals, ramification, discriminants, and the ideal class group, taking efforts to illustrate these results with calculations from accessible classes of examples such as quadratic, cubic, and cyclotomic fields. We then bring Galois theory into the discussion and study the Galois action on ideals and primes: we construct the decomposition and inertia subgroups and study the action of Frobenius. Finally, we bring in additional tools from geometry (lattices and the geometry of numbers) and analysis (the zeta function and L-functions) to extend our results further: we use Minkowski's theorem to establish the finiteness of the ideal class group and to prove Dirichlet's Unit Theorem, and we establish the analytic class number formula and study the distribution of primes.
Time permitting, we may also discuss other topics according to student and instructor interest, such as p-adic valuations and completions, local fields, ray class fields, an introduction to class field theory, or other related topics. |
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| Textbook: I will write course notes (see below) in lieu of an official textbook, since I will be drawing material from a variety of sources. The presentation will parallel most closely the discussions in Marcus's "Number Fields" and Narkiewicz's "Elementary and Analytic Theory of Algebraic Numbers". Another good reference on these topics is Shafarevich and Borevich's "Number Theory", and for the more advanced with algebraic-geometry background, Neukirch's "Algebraic Number Theory". | |||||
| Background: There are no formal prerequisites, but students should have comfort with algebra at the beginning graduate level (Math 5111 or 5112). I will freely refer to some results from commutative algebra, and potentially complex analysis, but the goal is to make the course as self-contained as possible. | |||||
| Grades: Grades will be based on attendance and participation (50%), and on occasional homework assignments (50%), which will be collaborative and may involve roundtable discussions of the solutions. Depending on student and instructor interest, there may also be student presentations on course-related topics during the semester. | |||||
| Lecture Notes |
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Current version of the lecture notes (through Lecture 8 on 9/19): Notes, Lecture 8 |
| Homework Assignments |
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Homework assignments will be posted here. There will be one assignment every 2-3 weeks for a total of 5 during the semester. A suggested number of problems to aim for solving on each assignment appears at the top of the assignment. You are not expected to solve all of the problems, although it is certainly worth your time to try to solve all of them.
Homework #1, due Mon Sep 23rd. Prepare to present 1-2 problems in class on the due date. |