  Our main result is a construction of four families C_1,C_2,B_1,B_2 which are
equipollent with the power set of the real line R and satisfy the following
properties. (i) The members of the families are proper subfields of R whose
algebraic closures equal the field C. (ii) Each field in C_1vC_2 contains a
Cantor set. (iii) Each field in B_1vB_2 is a Bernstein set. (iv) All fields in
C_1vB_1 are isomorphic. (v) If K,L are fields in C_2vB_2 then K is isomorphic
to a subfield of L only in the trivial case K=L.

Commutative Algebra;Logic