On the cohomological dimension of non-standard number fields

Abstract

The study made in this paper was motivated by the following question: given n 2 0, is the class of fields of cohomological dimension in elementary? It is relatively easy to see that this class is closed under elementary substructures, see Proposition 2.5; for n12 it is however unknown whether it is closed under ultraproducts, or even ultrapowers. Klingen showed in [6] that the class of fields having a class formation is elementary. As fields having a class formation have cohomological dimension 2, this suggests the following type of question: given an elementary class generated by fields of cohomological dimension 5 2, do all its elements have cohomological dimension 12? In this paper we propose to show that the elementary class S generated by all (totally imaginary) global fields and their algebraic extensions contains only fields of cohomological dimension ~2 (Theorem 3.6). This implies in particular that cd(Q*(i)) =2. We also study, for Q* a non-standard extension of Q, some of the properties of G(Q*); in contrast with the standard case, G(Q*) contains non-pro-cyclic abelian subgroups (Proposition 3.7), and cd(Q*(p))> 1 (Corollary 3.8). However, as in the standard case, we have cd(Q*“b)= 1 (Theorem 3.11). Our study allows us, given a prime number p, to find a sentence ulp such that for every field KES