Stable division rings

It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings. Macintyre proved any ω-stable field to be either finite or algebraically closed [7]. This was generalised by Cherlin and Shelah to superstable fields [3]. It follows that a superstable division ring is a field [2]. The result was broadened to supersimple division rings by Pillay, Scanlon and Wagner in [8]. As for stable fields, infinite ones are conjectured to be separably closed. Scanlon proved that an infinite stable field has no Artin-Schreier extension [10]. Wagner adapted the argument to show that a simple field has only finitely many Artin-Schreier extensions [6]. Proving commutativity usually goes in two steps, showing first that the ring viewed as a vector space over its centre must have finite dimension, and proving that the centre cannot have skew extensions of finite degree. Concerning a stable division ring, at least can we show that in positive characteristic, it must have finite dimension over its centre. This also holds for a simple division ring. 1. One word on stable structures In a given theory T , a formula f(x, y) is said to have the order property if it totally orders an infinite sequence, i.e. if there exists an infinite sequence a1, a2 . . . such that T |= f(ai, aj) if and only if i < j The formula f has the strict order property if it defines a partial ordering with infinite chains, i.e. if there exists an infinite sequence a1, a2 . . . such that T |= ∧ i<j f(ai, aj) ∧ ai 6= aj If a formula has the strict order property, it has the order property. Definition 1. A theory is stable if no formula has the order property. A structure is stable if its theory is so. We refer to [9] and [12] for details about stable groups. We just recall that to any formula f(x, y) in a group without the strict order property is associated an integer n, such that any strictly decreasing chain of subgroups defined by formulae f(x, a1), . . . , f(x, am) have no more than n elements. Moreover : 2000 Mathematics Subject Classification. 03C45, 03C60, 16K20.