Mathematics MS Thesis Defense by Çağatay Altuntaş

KOÇ UNIVERSITY

GRADUATE SCHOOL OF SCIENCES & ENGINEERING

MATHEMATICS

MS THESIS DEFENSE BY ÇAĞATAY ALTUNTAŞ

Title: Dedekind Zeta Function and Class Number Formula.

Speaker: Çağatay Altuntaş

Time: August 28, 2018, 17:00

Place: ENG 208

Koç University

Rumeli Feneri Yolu

Sariyer, Istanbul

Thesis Committee Members:

Assoc. Prof. Kazım Büyükboduk (Advisor, Koç University)

Prof. Tolga Etgü (Koç University)

Assoc. Prof. Ayhan Günaydın (Boğaziçi University)

Abstract:

In this thesis, we first introduce number fields and their ring of integers. We first show that the ring of integers of a number field is a Dedekind Domain. To study number fields in details, we present a geometric approach and show that the class number of a number field is finite and characterize the group of units of a ring of integers. After that, we introduce the Dedekind zeta function of a number field. Moreover, we present the Analytic Class Number Formula which states that the Dedekind zeta function converges for any complex number with real part greater than 1 and has a simple pole at the point 1. We show that its residue at 1 is given by particular invariants of a number field. Lastly, we present various arguments to evaluate class number of number fields.