## Algebraic Number Theory I

• Winter semester 2003/2004
• Lecturer: Y. Sommerhäuser
• Time: Wednesday, 9:15 am
• Room: 251
• Contents: Number theory is one of the largest mathematical fields, whose origins reach back to ancient times, but on the other hand is in the focus of current research. The part of number theory that is called "algebraic" was developed in the nineteenth and the beginning twentieth century by mathematicians like E. Kummer, R. Dedekind, L. Kronecker, D. Hilbert, K. Hensel and H. Hasse, to name just a few. In the first semester, we will develop the ideal-theoretic approach of Dedekind. We begin with the notion of an algebraic integer in a finite extension of the rational numbers and show that these numbers form a ring. In these rings, as it was first noticed by E. Kummer, there is in general no unique factorization into prime numbers, but instead we have a unique factorization into prime ideals. Such rings are called Dedekind rings. We discuss the ideal class group, show how prime numbers and ideals behave under the embedding into a larger ring, prove Minkowski's theorem and dicuss discriminants and differents. The whole theory can excellently be illustrated through easily accessible examples, of which we will discuss many explicitly. The literature on algebraic number theory is very extensive - only a brief selection is given below. We will follow closely the book of J. Neukirch cited there.
• References:
• S. Borewicz/I. Safarevic: Zahlentheorie, Birkhäuser, Basel, 1966
• H. Hasse: Zahlentheorie, Akademie-Verlag, Berlin, 1969
• H. Koch: Zahlentheorie, Vieweg, Braunschweig, 1997
• S. Lang: Algebraic Number Theory, Grad. Texts Math., Bd. 110, Springer, Berlin, 1994
• W. Narciewicz: Elementary and Analytic Theory of Algebraic Numbers, 2. Aufl., Springer, Berlin, 1990
• J. Neukirch: Algebraische Zahlentheorie, Springer, Berlin, 1992