On nonelementarily equivalent pairs of fields (Q1400607)

Let \(T\) be the complete theory of the field of rational numbers. If \(Q\) is a model of \(T\), let \(Q^r\) be its real closure. The paper gives an answer to the following question of Victor Harnik: Are there nonstandard models \(Q_1\) and \(Q_2\) of \(T\) such that \((Q_1^r,Q_1)\) and \((Q_2^r,Q_2)\) are elementarily inequivalent? The positive answer is given in two steps. In the first, it is shown that the standard natural numbers \(\mathbb{N}\) are uniformly definable in structures of the form \((Q^r,Q)\), where \(Q\) is a model of \(T\). In the second step, an example involving coding the satisfaction relation on \(({\mathbb{N}},+,\times)\) is constructed to show that there are nonstandard elementary extensions of \(({\mathbb{N}},+,\times)\), \(N_1\) and \(N_2\), such that \((N_1,{\mathbb{N}})\) and \((N_1,{\mathbb{N}})\) are elementarily inequivalent. There is also an outline of the argument showing how to obtain continuum many pairwise inequivalent structures.