Number Theory 3

Course at FU Berlin, Winter Term 14/15

Kay Rülling


Class field theory is one of the high peaks in the development of number theory in the first half of the 20th century.
It aims at understanding the finite Galois extensions with abelian Galois group of a number field K (i.e. a finite extension of the rational numbers).
One application is for example the theorem of Kronecker-Weber stating that the finite abelian Galois extensions of the rational numbers are
exactly those extensions, which are contained in a cyclotomic field extension.

In order to obtain the description for a number field, which is considered to be a global field, one first proves a local version, involving local fields.
These fields arise by completing a number field along its various primes. By local class field theory the abelian Galois extensions of a
local field L correspond to certain subgroups of L\{0}. In the course we will discuss infinite Galois theory, global and local fields and
give the formulation of global and local class field theory. We will give the proofs for local class field theory in some detail and discuss the global version
as time permits.

Prerequisites: Basic knowledge in (finite) Galois theory, commutative algebra and number theory.


Lecturer Time Room
Lecture Kay Rülling Mo 10-12 SR025/026/A6
Exercise Lars Kindler Fr 10-12 SR009/A6

The first lecture is on October 13, 2014. The first exercise session is on October 24.
If you have any questions concerning the lectures or the exercises feel free to contact one of us.


There will be an oral exam. Please write an email to Kay Rülling to fix a date.


Every Monday there will be a new exercise sheet, the solutions are discussed in the exercise session on the Friday of the following week.

  1. Exercise sheet October 13, 2014.
  2. Exercise sheet October 20, 2014 (ver 2, in Ex 2.4(2) H is now profinite).
  3. Exercise sheet October 27, 2014.
  4. Exercise sheet November 03, 2014 (ver 2, formulation of Ex 4.2(3) corrected).
  5. Exercise sheet November 10, 2014.
  6. Exercise sheet November 17, 2014.
  7. Exercise sheet November 24, 2014.
  8. Exercise sheet December 1, 2014(ver 2, added char(k) not 2 in Ex 8.3).
  9. Exercise sheet December 8, 2014 (ver 2, Ex 9.3.1 corrected).
  10. Exercise sheet December 15, 2014.
  11. Exercise sheet January 5, 2015 (ver 2, Ex 11.2(1) corrected).
  12. Exercise sheet January 12, 2015 (ver 3, various typos fixed).
  13. Exercise sheet January 19, 2015.


Here is a list of references for the course:

  1. K. Kato, N. Kurokawa, T. Saito, Number Theory 2, Iwanami Series in Modern Mathematics, vol 240
  2. J. Neukirch, Algebraische Zahlentheorie , Springer
  3. J.-P. Serre, Local fields, Graduate Texts in Mathematics, vol 67, Springer
  4. J.S. Milne, Class Field Theory (v4.02), 2013. Available at
  5. J.W.S Cassels, A. Fröhlich (Ed.), Algebraic Number Theory, London Mathematical Society, 2010.
  6. J.C. Jantzen, J. Schwermer, Algebra , Springer, 2006.
  7. N. Jacobson, Basic Algebra II, W.H. Freeman and Company, 1980.