For a number field K over Q, there is an associated invariant called the class number, which captures how far the ring of integers of K, is from being a principal ideal domain (PID). The study of class numbers is an important theme in algebraic number theory. In order to understand how the class number varies on varying the number field, Siegel showed that the class number times the regulator approaches infinity for a sequence of quadratic number fields. Later, Brauer extended this to a sequence of Galois extensions over Q, with some additional hypothesis. This is called the famous Brauer-Siegel Theorem. Recently, Tsfasman and Vladut conjectured a generalized Brauer-Siegel statement for sequence of number fields. In this talk, we prove the classical Brauer-Siegel and the generalized version in several unknown cases.
Seminar Hall, SMS
University of Toronto
On the generalized Brauer-Siegel Theorem