On Finding Fields from Their Algebraic Closure Geometries

It is shown that if F\ and Fi are algebraically closed fields of nonzero characteristic p and F\ is not isomorphic to a subfield of F2 , then F\ does not embed in the skew field of quotients 0Fj of the ring of morphisms of the additive group of F2 . From this fact and results of Evans and Hrushovski, it is deduced that the algebraic closure geometries G(K¡/Fi) and (7(^2/^2) are isomorphic if and only if K\ : F\ ~ K% : F2 • It is further proved that if Fq is the prime algebraically closed field of characteristic p and F has positive transcendence degree over Eg , then Op and Of0 are not elementarily equivalent. 0. Introduction If F and K are algebraically closed fields with F a subfield of K, the set of algebraically closed extensions F' of F in K with t.d.(F'/F) = 1 forms a geometry under algebraic dependence. We denote this geometry by G(K/F), and impose the minor restriction that t.d.(K/F) > 3 to avoid trivialities. Question. If F', F" , K', K" \= acfp , when is G(K'/F') ~ G(K"/F")1 Since t.d.(K/F) 1 is the dimension of the geometry G(K/F), it is clearly necessary that t.d.(K'/F') = t.d.(K"/F") if the geometries are to be isomorphic. We will show that, indeed, F' ~ F" as well, by showing that if t.d.(F"/Fp) > t.d.(F'/Fp), then there are projective planes of G(K"/F") that are not isomorphic to planes of G(K'/F'). (The notion of "projective plane of will be clarified below.) The proof of F' ~ F" in the case p ^ 0 will be based upon a partial study of the skew field of quotients, cfp , of the ring of /A-polynomials in one variable over F. (The approach of distinguishing G(K'/F') and G(K"/F") by studying the cfF was suggested by Evans and Hrushovski in [2].) We will also Received by the editors December 5, 1990 and, in revised form, April 17, 1991. 1991 Mathematics Subject Classification. Primary 03C60; Secondary 12L12.