Submission date: 1st July 2022

Abstract:

We show that the ring of integers of ℚ^{tr} is existentially definable in the ring of integers of ℚ^{tr}(i), where ℚ^{tr} denotes the field of all totally real numbers. This implies that the ring of integers of ℚ^{tr}(i) is undecidable and first-order non-definable in ℚ^{tr}(i). More generally, when L is a totally imaginary quadratic extension of a totally real field K, we use the unit groups R^× of orders R ⊆ O_L to produce existentially definable totally real subsets X ⊆ O_L. Under certain conditions on K, including the so-called JR-number of O_K being the minimal value JR(O_K) = 4, we deduce the undecidability of O_L. This extends previous work which proved an analogous result in the opposite case JR(O_K) = ∞. In particular, unlike prior work, we do not require that L contains only finitely many roots of unity.

Mathematics Subject Classification: 11U05

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Full text arXiv 2207.00140: pdf, ps.