Algebraic Number Theory 2

This course is a continuation of the course Algebraic Number Theory 1 The main topic is the theory of the field of $p$-adic numbers and its finite extensions, and in particular local class field theory which gives a description of the maximal abelian quotient $\mathop{\rm Gal}(K^{\rm sep}/K)^{\rm ab}$ of such a field $K$ in terms of “$K$ itself”. Hopefully at the end there will be enough time left to sketch the implications of this for number fields.

Prerequisites: Linear Algebra and Algebra. Of course it is an advantage if you attended the Algebraic Number Theory 1 course, but this is not absolutely required. You do need some basic knowledge about fields with non-archimedean valuations however. Feel free to contact me if you need assistance with learning these things on your own.

Dates: Tue, Fri, 10-12 (First lecture: April 12). Place: S-U-3.02.

Exercise group: (Dr. F. Fité) Wed, 12-2pm, S-3.14.

Problem sheets

Sheet 1 pdf April 26, 2016
Sheet 2 pdf May 3, 2016
Sheet 3 pdf May 10, 2016
Sheet 4 pdf May 17, 2016
Sheet 5 pdf May 24, 2016
Sheet 6 pdf May 31, 2016
Sheet 7 pdf June 7, 2016
Sheet 8 pdf June 14, 2016
Sheet 9 pdf June 21, 2016
Sheet 10 pdf June 28, 2016
Sheet 11 pdf July 5, 2016
Sheet 12 pdf July 12, 2016

Content of the course

Here is a rough outline of what we will cover:

  • Hensel’s Lemma and consequences. Unramified, tamely ramfied, wildly ramified extensions of local fields.
  • (if necessary) Infinite Galois theory
  • Group Cohomology
  • Local Class Field Theory
  • Formal groups, Lubin-Tate theory
  • (hopefully) Statement of the main results of Global Class Field Theory


For large parts of the course, I will follow Milne’s notes:

Further references that might be useful are

  • Cassels, Fröhlich (eds.), Algebraic Number Theory
  • Neukirch, Klassenkörpertheorie /Class field theory
  • Neukirch, Zahlentheorie/Algebraic Number theory
  • Artin, Tate, Class field theory
  • T. Yoshida, Local class field theory via Lubin-Tate theory, Annales de la Faculte des Sciences de Toulouse, Ser. 6, 17-2 (2008), 411-438. math.NT/0606108
  • Serre, Local fields / Corps locaux
  • Fesenko, Vostokov, Local fields and their extensions, AMS Transl. of Math. Monographs 121

Survey articles on class field theory and further developments: