Classically projective groups and pseudo classically closed fields

Introduction Let K be a PAC (pseudo algebraically closed) field. Then the absolute Galois group GK of K is projective by Ax [A], i.e., group extensions of GK by a finite groups H \Gamma! GK ! 1 are split: There exists oe : GK ! F such that oe = id GK . On the other hand, Gruenberg [G] showed that for a profinite projective group G, all group extensions of G by profinite groups H are split. Using Gruenberg's theorem Lubotzky -- van den Dries [L--vdD] solved the inverse absolute Galois problem for projective groups as follows: 2000 Mathematics Subject Classification. Primary 11, 12, 14; Secondary 11: G, S20, U09; 12: D, E20, F, L; 14: E, G. c fl0000 American Mathematical Society For projective profinte groups G there exist PAC fields K with GK = G. Extending the category of PAC fields, one introduced the PRC (pseudo real closed) fields, the PpC (pseudo p-adically closed) fields and further, fields regularly closed with respect to finitely many henselisations and orderings, se