Implicit Definability of Subfields

Abstract

We say that a subset A of M is implicitly definable in M if there exists a sentence $\varphi(P)$ in the language $\mathcal{L}(M) \cup \{P\}$ such that A is the unique set with $(M,A) \models \varphi(P)$. We consider implicit definability of subfields of a given field. Among others, we prove the following: $\overline{\mathbb{Q}}$ is not implicitly $\emptyset$-definable in any of its (proper) elementary extension $K \succ \overline{\mathbb{Q}}$. $\mathbb{Q}$ is implicitly $\emptyset$-definable in any field K (of characteristic 0) with tr.deg $_{\mathbb{Q}}K < \omega$. In a field extension $\mathbb{Q} < K$ with K algebraically closed, $\mathbb{Q}$ is implicitly definable in K if and only if tr.deg $_{\mathbb{Q}}(K)$ is finite.