18.785 | Fall 2021 | Graduate

Number Theory I


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session


This is the first semester of a one-year graduate course in number theory. There is no official prerequisite.


18.705 Commutative Algebra

More on Prerequisites

Die Mathematik ist die Königin der Wissenschaften und die Zahlentheorie ist die Königin der Mathematik.” —Carl Friedrich Gauss  

[Translation: “Mathematics is the queen of the sciences and number theory is the queen of mathematics.”]

As suggested by this quote, number theory is supported by many subfields of mathematics, and we will not hesitate to call upon them as needed. In most cases these supporting subjects will play a minor role, but you should be aware that at various points in the course we will make reference to standard material from many other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry. When this happens, I will include in the lecture notes a quick review of any terminology and theorems we need that fall outside of the official corequisite for this course, which is 18.705 Commutative Algebra. Note that 18.705 transitively includes 18.100C Real Analysis, as well as 18.701 Algebra I and 18.702 Algebra II, as prerequisites. In past years, 18.112 Complex Analysis was also a formal prerequisite. This is no longer the case, but if you have never studied complex analysis you will need to be prepared to do some extra reading when we begin our study of zeta functions and L-functions.

For graduate students in mathematics (the target audience of this course), none of this should be an issue. Undergraduates and students from other departments may need to spend some time acquainting (or reacquainting) themselves with supporting material as it arises.

Motivated undergraduate students with adequate preparation are welcome to register for this course, but should do so with the understanding that it is a graduate-level course aimed at students who are planning to do research in number theory or a closely related field. I expect students taking this course to be amply motivated and to take personal responsibility for mastering the material—this includes doing whatever outside reading may be necessary to fill in any gaps in your background.

Course Overview

Historically, number theory has often been separated into algebraic and analytic tracks, but we will not make such a sharp distinction. Indeed, one of the central themes of modern number theory is the intimate connection between its algebraic and analytic aspects. These connections lie at the heart of many of recent breakthoughs and current areas of research, including the modularity theorem, the Sato-Tate theorem, the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, and the Langlands program.

Having said that, number theory is, at its core, the study of numbers. Our starting point is thus the integer ring Z, its field of fractions Q, and the various completions and algebraic extensions of Q. This means we will initially cover many of the standard topics in algebraic number theory, including Dedekind domains, decomposition of prime ideals, local fields, ramification, the discriminant and different, ideal class groups, and Dirichlet’s unit theorm. We will spend roughly the first half of the semester on these topics, and then move on to some closely related analytic topics, including zeta functions and L-functions, the prime number theorem, primes in arithmetic progressions, the analytic class number formula, and the Chebotarev density theorem. We will also present the main theorems of local and global class field theory (one of the crowning achievements of early 20\(^{\text{th}}\)-century number theory), but we will only have time to cover a few parts of the proofs.

Textbooks and Lecture Notes

There is no required textbook. Lecture notes will be provided.

You are encouraged to take notes in class that include definitions and statements of lemmas and theorems, but only a high-level summary of the proofs (many of which I will only sketch in class in any case). After class, you should attempt to fill in the proofs on your own. This is a great way to learn and will help you absorb the material much more effectively than a purely passive approach. You can then consult the lecture notes I will provide and/or any of the texts below to fill in gaps and to compare your approach with mine.

Number theory is a vast subject, and it is good to see it from many different perspectives. Below are a number of standard references that I can recommend. The classic text of Cassels and Frohlich is not “officially” available in online form, but googling “Cassels and Frohlich” will quickly lead you to several scanned versions; alternatively, you can purchase the 2010 reprint by the London Mathematical Society, which corrects most of the errata noted below.

Suggested Textbooks

Altman, Allen, and Steven Kleiman. A Term of Commutative Algebra. Worldwide Center of Mathematics, 2013. ISBN: 9780988557215. (errata).

Cassels, John William Scott, and Albrecht Fröhlich, eds. Algebraic Number Theory. 2\(^{\text{nd}}\) edition. London Mathematical Society, 2010. ISBN: 9780950273426. (This resource may not render correctly in a screen reader.errata (PDF)).

Davenport, Harold. Multiplicative Number Theory. 3rd edition. Springer, 2000. ISBN: 9780387950976.

———. Class Field Theory, 2013. Available at J. S. Milne Mathematics Site.

Lang, Serge. Algebraic Number Theory. 2nd edition. Springer, 2013. ISBN: 9781461269229.

Lorenzini, Dino. An Invitation to Arithmetic Geometry. American Mathematical Society, 1996. ISBN: 9780821802670.

Manin, Yuri Ivanovic, and Alexei A. Panchishkin. Introduction to Modern Number Theory. 2nd edition. Springer, 2007. ISBN: 9783540203643.

Milne, J. S. Algebraic Number Theory, 2017. Available at J. S. Milne Mathematics Site.

Neukirch, Jürgen. Algebraic Number Theory. Springer, 2010. ISBN: 9783642084737.

Rosen, Michael. Number Theory in Function Fields. Springer, 2010. ISBN: 9781441929549.

Serre, Jean-Pierre. A Course in Arithmetic. Springer, 1973. ISBN: 9780387900414.

———. Local Fields. Springer, 2013. ISBN: 9781475756753. [Preview with Google Books]

Other Useful Texts

I can also recommend the following texts, according to taste (Atiyah-MacDonald is an examplar of brevity, while Eisenbud is wonderfully discursive; Matsumura, my personal favorite, is somewhere in between, and I also recommend Milne’s primer):

Atiyah, M. F., and I. G. MacDonald. Introduction to Commutative Algebra. CRC Press*, 1994. ISBN: 9780201407518.*

Eisenbud, David. Commutative Algebra with a View Toward Algebraic Geometry. Springer, 2008. ISBN: 9780387942698.

Matsumura, Hideyuki. Commutative Ring Theory. Cambridge University Press, 1989. ISBN: 9780521367646. [Preview with Google Books]

Milne, J. S. This resource may not render correctly in a screen reader.A Primer of Commutative Algebra (PDF), 2017. Available at J. S. Milne Mathematics Site.

Problem Sets

Weekly problem sets will be posted online. Solutions are to be prepared in typeset form (typically via LaTex) and submitted electronically as a pdf file by noon on the due date. Late problem sets will not be accepted—your lowest score is dropped, so you can afford to skip one problem set without penalty. Collaboration is permitted/encouraged, but you must write up your own solutions and explicity identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing Sources consulted: none at the top of your submission.


Your grade will be determined by your performance on the problem sets; your lowest score will be ignored (this includes any problem set you did not submit). There are no exams and no final.

Course Info

As Taught In
Fall 2021
Learning Resource Types
Online Textbook
Lecture Notes
Problem Sets