Notre Dame Journal of Formal Logic

Implicit Definability of Subfields

Kenji Fukuzaki and Akito Tsuboi


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.

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Notre Dame J. Formal Logic, Volume 44, Number 4 (2003), 217-225.

First available in Project Euclid: 29 July 2004

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Primary: 03C40: Interpolation, preservation, definability

implicit definability fields


Fukuzaki, Kenji; Tsuboi, Akito. Implicit Definability of Subfields. Notre Dame J. Formal Logic 44 (2003), no. 4, 217--225. doi:10.1305/ndjfl/1091122499.

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