Winter 2017, Section 1

Math 776, as the continuation of Math 676 , is a second-semester graduate course in algebraic number theory. While Math 676 covered a variety of basic topics in algebraic number theory,
Math 776 traditionally focuses on a single topic: Class Field Theory, the study of abelian extensions of number fields.

Course details

Time: Tuesday-Thursday 2:30pm-4:00pm
Place: 1866 East Hall
First meeting: Thursday, January 5, 2017

About the instructor

Jeffrey Lagarias
Office: 3086 East Hall
Office Phone: 763-1186
Office Hours: TBD or by appointment.

Course description

The plan for this course is under construction, to be discussed on the first day of class. This concerns to what extent the course will cover material
relevant to understanding parts of S. Mochizuki's IUT theory. A user-friendly approach to that theory would use the text

  • J. Neukirch, Class Field Theory [CFT], Grundlehren der Mathematischen Wissenschaften, 280, Springer-Verlag, Berlin 1986

    This book is different from Neukirch's other books on algebraic number theory and class field theory. It give an abstract version of class field theory,
    encoding it as purely group-theoretic constructions, and treats infinite extensions of algebraic number fields, starting at page 1.
    It gets to the main results quickly, but is opaque if you never saw class field theory before, and even if you have.
    I am considering to try it.

    Other Textbooks

    A primary text I have used before is

    Serge Lang,Algebraic Number Theory, 2nd Edition Springer-Verlag 1994.

    This text seems a bit dry and unmotivated, but it covers the right things on the analytic side.

    A second very
    good source is J.S. Milne's course notes on class field theory, which can be downloaded here.

    A classical source is

    G.Janusz, Algebraic Number Fields, 2nd Edition (AMS 2005)

    A very good modern source is:

    J. Neukirch, Algebraic Number Theory [ANT] (Springer-Verlag 1999).

    Students are encouraged to have at least one of these last two sources available.

    Previous Versions

    The following summarizes previous versions of the course.

    Class field theory, the study of abelian extensions of number fields, was a crowning achievement of number theory in the first half of the 20th century. It brings together, in a unified fashion, the quadratic and higher reciprocity laws of Gauss, Legendre et al, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.

    This approach in this course will be to begin with review and then to prove the Kronecker-Weber theorem (class field theory over the rational field). Then we cover Kummer theory, the formulations of the statements of global class field theory, for number fields, and local class field theory, for p-adic fields. Then it does other stuff.

    Possible Topics


    Math 676 or equivalent. Graduate coursework in algebra is also recommended. It would help (me) if you know more algebra than I know. If you are unsure how your background matches these prerequisites, see me.
    Jeff Lagarias (lagarias(at)