Division probability for quaternion algebras over fields

Abstract

Let [Formula: see text] be a field of [Formula: see text] with finitely many square classes. In this paper, we define a new rational valued invariant of [Formula: see text], and call it the division probability of [Formula: see text]. We compute it for all fields of elementary type. Further, we show that [Formula: see text], where [Formula: see text] is the number of Witt-equivalence classes of fields with [Formula: see text], and [Formula: see text] is the count of rational numbers that appear as division probabilities for fields [Formula: see text] of elementary type with [Formula: see text]. In the paper, we also determine [Formula: see text] for all [Formula: see text] and show that rational numbers of type [Formula: see text] always occur as division probability for a suitable field [Formula: see text].