Small skew fields

A division ring of positive characteristic with countably many pure types is a field. Wedderburn showed in 1905 that finite fields are commutative. As for infinite fields, we know that superstable [1, Cherlin, Shelah] and supersimple [4, Pillay, Scanlon, Wagner] ones are commutative. In their proof, Cherlin and Shelah use the fact that a superstable field is algebraically closed. Wagner showed that a small field is algebraically closed [5], and asked whether a small field should be commutative. We shall answer this question positively in non-zero characteristic. 1. Preliminaries Definition 1.1. A theory is small if it has countably many n-types without parameters for all integer n. A structure is small if its theory is so. We shall denote dcl(A) the definable closure of a set A. Note that if K is a field and A a subset of K, then dcl(A) is a field too. Smallness is clearly preserved under interpretation and addition of finitely many parameters. Let D,D1, D2 be A-definable sets in some structure M with A ⊂M . We define the Cantor-Bendixson rank CBA(D) and degree dCBA(D) of D over A. Definition 1.2. By induction, we define CBA(D) ≥ 0 if D is not empty CBA(D) ≥ α+1 if there is an infinite family of disjoint A-definable subsets Di of D, such that CBA(Di) ≥ α for all i < ω. CBA(D) ≥ β for a limit ordinal β if CBA(D) ≥ α for all α < β. Definition 1.3. dCBA(D) is the greatest integer d such that D can be divided into d disjoint A-definable sets, with same rank over A as D. Proposition 1.4. If M is small and A is a finite set, (i) The rank CBA(M) is ordinal. (ii) The degree dCBA is well defined. (iii) If D1 ⊂ D2, then CBA(D1) ≤ CBA(D2). (iv) CBA(D1 ∪D2) = max{CBA(D1), CBA(D2)}. (v) CBA and dCBA are preserved under A-definable bijections. If A is empty, we shall write CB and dCB rather than CB∅ or dCB∅. 2000 Mathematics Subject Classification. 03C15, 03C50, 03C60, 12E15.