Implicit Definability of Subfields

Notre Dame Journal of Formal Logic 44 (4):217-225 (2003)
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Abstract

We say that a subset A of M is implicitly definable in M if there exists a sentence $\phi$ in the language $\mathcal{L} \cup \{P\}$ such that A is the unique set with $ \models \phi$. 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 elementary extension $K \succ \overline{\mathbb{Q}}$. $\mathbb{Q}$ is implicitly $\emptyset$-definable in any field K 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 $_{\mathbb{Q}}$ is finite

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The undecidability of pure transcendental extensions of real fields.Raphael M. Robinson - 1964 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 10 (18):275-282.
The Undecidability of Algebraic Rings and Fields.Julia Robinson - 1964 - Journal of Symbolic Logic 29 (1):57-58.
Definability of Initial Segments.Akito Tsuboi & Saharon Shelah - 2002 - Notre Dame Journal of Formal Logic 43 (2):65-73.

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