Submission date: 19 October 2020

Abstract:

In this paper we investigate the algebraic extensions K of ℚ in which we cannot existentially or universally define the ring of integers O_K. A complete answer to this question would have important consequences. For example, the existence of an existential definition of ℤ in ℚ would imply that Hilbert's Tenth Problem for ℚ is undecidable, resolving one of the biggest open problems in the area. However, a conjecture of Mazur implies that the integers are not existentially definable in the rationals. Although proving that an existential definition of ℤ in ℚ does not exist appears to be out of reach right now, we show that when we consider all algebraic extensions of ℚ, this is the generally expected outcome. Namely, we prove that in most algebraic extensions of the rationals, the ring of integers is not existentially definable. To make this precise, we view the set of algebraic extensions of ℚ as a topological space homeomorphic to Cantor space. In this light, the set of fields which have an existentially definable ring of integers is a meager set, i.e. is very small. On the other hand, by work of Koenigsmann and Park, it is possible to give a universal definition of the ring of integers in finite extensions of the rationals, i.e. in number fields. Still, we show that their results do not extend to most algebraic infinite extensions: the set of algebraic extensions of ℚ in which the ring of integers is universally definable is also a meager set

Mathematics Subject Classification: 11U05, 12L05, 11U09, 03D45

Keywords and phrases:

Full text arXiv 2010.09551: pdf, ps.