On nonelementarily equivalent pairs of fields

In [3], Keisler proves that if Q1 is a nonstandard model of the complete theory Th(Q) of (Q;+;×; 0; 1) and C1 is the algebraic closure of Q1, then (C1; Q1) is elementarily equivalent to (C;Q) where C is the algebraic closure of Q. But if Q r 1 is the real closure of Q1 and Qr the real closure of Q, then it is observed in [1] that (Q r 1; Q1) ≡(Qr ;Q). Victor Harnik asked the following question: if we have two nonstandard models Q1 and Q2 of Th(Q) are (Q r 1; Q1) and (Q r 2; Q2) elementarily equivalent? Let L be the language {0; 1;+;×} and let N be a nonstandard model of Peano Arithmetic, formulated in L. We refer to “formulas” of L meaning all the elements of N satisfying a formula F(x), naturally interpreting in the standard model N “x is the G< odel number of a formula”. Then “standard formula” means a “formula” which belongs to the standard part of N (i.e. N) and hence can be identi$ed with a real formula. We call “standard formulas with arbitrary parameters” all the “formulas” of the form G(b) where G(x) is a “formula” with one free variable with a standard G< odel number and b is an integer (standard or nonstandard).