Implicit Definability of Subfields

Implicit Definability of Subfields

Kenji Fukuzaki and Akito Tsuboi

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

Abstract

We say that a subset A of M is implicitly definable in M if there exists a sentence $\phi(P)$ in the language $\mathcal{L}(M) \cup \{P\}$ such that A is the unique set with $(M,A) \models \phi(P)$ . 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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Primary Subjects: 03C40

Keywords: implicit definability; fields

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1091122499
Digital Object Identifier: doi:10.1305/ndjfl/1091122499
Mathematical Reviews number (MathSciNet): MR2130307
Zentralblatt MATH identifier: 02186738

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