18.786 | Spring 2016 | Graduate

Number Theory II: Class Field Theory


Course Reading List

There is no required textbook, but lecture notes are provided. An annotated bibliography for this course is listed below.

Quadratic Extensions of Local and Global Fields

  • Serre, Jean-Pierre. A Course in Arithmetic. New York: Springer, 2001. ISBN: 9780387900407.
        The Hilbert symbol is the main subject of Part I of this book, especially Chapter III. It only treats completions of Q, but the material generalizes away from Q2 to general local fields. The book emphasizes elementary techniques, so can give a feeling for what is going on in a more abstract framework.

Local Class Field Theory

  • Serre, Jean-Pierre. Local Fields. Vol. 67. New York, NY: Springer, 2013. ISBN: 9781475756739.
        A classic reference that rewards the effort you put into it. It begins with the structure theory of local fields, develops group cohomology from scratch, and then proves the main theorem of local class field theory. Unfortunately, this book does not do the work of plainly laying bare its main threads, so requires some patience for self-study.
  • Fesenko, I. B., and S. V. Vostokov. Local Fields and Their Extensions. Providence, RI: American Mathematical Society, 2002. ISBN: 9780821832592.
        A newer reference, with updates on the developments of the subject since Serre. Very detailed, with many exercises. An online version (PDF) is also available.

Class Field Theory (Local and Global)

  • Artin, Emil, and John Torrence Tate. Class Field Theory. Vol. 366. American Mathematical Society, 1967.
        An original source for many of the ideas of global class field theory. Unfortunately, it does not treat local class field theory.
  • Cassels, J. W. S., and A. Fröhlich. Algebraic Number Theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO advanced study institute) with the support of the International Mathematical Union. New York, NY: Academic Press, 1967. ISBN: 9780121632519.
        Notes from an instructional conference, developing the whole theory more or less from scratch. Notes available from many different authors. The quality sometimes varies, but is often high. In particular, Serre and Tate contribute the notes on local and global class field theory respectively, and generally speaking, anything written by either of them is required reading.
  • Weil, André. Basic Number Theory. Vol. 144. New York, NY: Springer, 2013. ISBN: 9781461298366.
        Another classic text. It gives a cohomological treatment of class field theory without every saying the words, which is both a bug and a feature. It is the only source I know with a detailed approach to the proof of the main global theorem via zeta functions.
  • Milne, James. Class Field Theory (PDF)
        These pleasantly written notes, which cover the subject in detail, are a solid reference for most of the ideas of class field theory.

Texts by Neukirch

Neukirch, who was an exemplary expositor, wrote two books with the same name:

  • Neukirch, Jürgen. Class Field Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986.
  • Neukirch, Jürgen. Class Field Theory: The Bonn Lectures. Heidelberg: Springer, 2013. ISBN: 9783642354366.
        The former is really a geodesic approach to the subject that minimizes the role of group cohomology, only using it in the case of cyclic groups, where it is more elementary.

Two other relevant books, one less advanced and one more advanced than the present course:

  • Neukirch, Jürgen. Algebraic Number Theory. Vol. 322. Berlin: Springer, 2010. ISBN: 9783642084737.
        This is a great introduction for the general background on number fields.
  • Neukirch, Jürgen, Alexander Schmidt, and Kay Wingberg. Cohomology of Number Fields. 2013. ISBN: 9783540378884.
        This is a more advanced treatment of Galois cohomology and its role in arithmetic.

Zeta Functions

  • Riemann, B. “On the Number of Primes Less Than a Given Magnitude.” Monthly Reports of the Berlin Academy. 1859.
        This is still the classic reference and definitely worth a read. English translations are readily available.
  • Tate, John Torrence. “Fourier Analysis in Number Fields and Hecke’s Zeta-functions.” PhD diss., Princeton University, Princeton, NJ, 1950.
        Tate’s thesis, which was reprinted in Cassels-Froöhlich. Its major contribution is to reinterpret Riemann’s work on the analytic properties of the zeta function by using Fourier analysis not on R/Z, but on AQ/Q. This is very useful for generalizing to number fields (c.f. to the treatment 2 of Hecke’s work in Neukirch’s Algebraic number theory), and much more clearly highlights the mechanisms underlying the analytic theory.

Homological Algebra

  • Gelfand, Sergei I. and Yuri I. Manin, Methods of Homological Algebra. Berlin: Springer, 2011. ISBN: 9783642078132.
        A great place to learn the subject. Different people tend to take different things away from it, which is a great sign of its richness. It gives a heavy emphasis on triangulated categories, and I personally think it would be improved by focusing more on chain complexes themselves.
  • Grothendieck, Alexandre. “Sur Quelques Points D’algèbre Homologique.” Tohoku Mathematical Journal, 2nd series, vol. 9, 1957: 119–183.
        There’s a lot that’s great about this paper still, but also a lot that’s dated. For example, I find the ∂-functors story to be dated (and uninspiring).
  • Cartan, Henri, and Samuel Eilenberg. Homological Algebra. Princeton (New Jersey): Princenton University Press, 1999. ISBN: 9780691049915.
        Another old text, with similar deficiencies. But it has many virtues, and is filled with examples. Much of the subject was invented here, which also makes it a great source from which to learn.

Course Info

As Taught In
Spring 2016
Learning Resource Types
notes Lecture Notes
assignment Problem Sets