Quadratic forms and a new invariant of fields

Abstract

In the thesis, for a field of characteristic different from 2 and finitely many square classes, a new rational valued numerical invariant has been introduced, namely the division probability of the field. It determines the probability that randomly picked quaternion algebra over the given field is division. Various results are given to compute this invariant for some well known fields having finitely many square classes. This invariant has given the characterization of generalized Hilbert fields (and generalized Hilbert schemes). If S is a Cordes scheme which is a finite product of Cordes schemes of local fields then this invariant determines the factors in the product. newline