Wonderful and nice extentions of number fields

Let F be a number field and A(F) be its adele ring. Consider the family I of all countable subfields of A(F),
containing F. I is closed under the unions of countable chains of elements of I. So in I there are existentially
closed (e-closed) elements.e-Closed (in right language) in I fields are the wonderful extentions of F. Its
elementary theory is complete and decidable and can be characterized explicitly (local-global principle LG, maximality
property M and etc.). Multiplicative group of a wonderful extention can be used (instead of idele group of F)
for a new formulation of global class field theory.

                                 References
1. Ershov Y.L. Subfields of the adeles ring. Algebra and logic,v.48,#6, pp418-...
2. Ershov Y.L. On wonderful extentions of the field of rational numbers. Doklady Mathematics,v.62,#1,2003,8-9
3. Ershov Y.L. Nice extentions and global class theory. Doklady Mathematics,v.67,#1,2003,21-23