Course on Algebraic Number Theory

Algebraic Number Theory is a fascinating area of mathematics, where the structure of the sets of natural numbers and of all integers, and natural generalizations of those, are studied by algebraic methods. While some of the underlying questions reach back several centuries, algebraic number theory is still an active and prospering area of research which has seen several spectacular new results and developments in the last few decades.

In the course, we will cover the basic theory of number fields, i.e., of the finite extension fields of $\mathbb Q$ (rings of integers, (prime) ideals, finiteness of the class group, …). Further information will follow later.

Dates: Tuesday 10-12 (S-U-3.02), Friday, 10-12 (S-U-3.01).

Prerequisites: Good knowledge of algebra (groups, rings and fields, field extensions, Galois theory).

Exercise group: There will be an exercise group lead by Dr. F. Fité, Friday, 8:30-10:00, S-U-3.02.

Contact: Ulrich Görtz,


There are many books on algebraic number theory. It might be best if you have a look at some of them yourself, in the library, to get an impression. A very incomplete list of books that are worth a look:

  • E. Hecke, Lectures on the Theory of Algebraic Numbers (originally in German, a real classic)
  • K. Ireland, M. Rosen, A classical introduction to modern number theory
  • G. Janusz, Algebraic Number Fields
  • D. Marcus, Number Fields
  • J. Milne, Algebraic Number Theory
  • J. Neukirch, Algebraische Zahlentheorie (also available in English)
  • P. Samuel, Théorie algébrique des nombres (also available in English; very inexpensive)
  • A. Schmidt, Einführung in die algebraische Zahlentheorie

Problem Sheets

Due date
Sheet 1 pdf Oct 27, 2015
Sheet 2 pdf Nov 3, 2015
Sheet 3 pdf Nov 10, 2015
Sheet 4 pdf Nov 17, 2015
Sheet 5 pdf Nov 24, 2015
Sheet 6 pdf Dec 1, 2015
Sheet 7 pdf Dec 8, 2015
Sheet 8 pdf Dec 15, 2015
Sheet 9 pdf Jan 12, 2016
Sheet 10 pdf Jan 19, 2016
Sheet 11 pdf Jan 26, 2016
Sheet 12 pdf Feb 2, 2016
Sheet 13 pdf Feb 9, 2016
Preparation for the exam pdf
Text of exam with solutions pdf
Text of make up exam pdf