Cantor sets and fields of reals

TL;DR

This paper constructs four large families of subfields of the real numbers with specific algebraic and topological properties, including containing Cantor and Bernstein sets, and explores their isomorphism relations.

Contribution

It introduces four families of subfields of R with prescribed algebraic and topological properties, expanding understanding of their structure and isomorphism relations.

Findings

Each family has cardinality equal to the power set of R.

Fields in C_1 and C_2 contain Cantor sets.

Fields in B_1 and B_2 are Bernstein sets.

Abstract

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.