Definability and decidability for rings of integers in totally imaginary fields [article]

Caleb Springer
2022 arXiv   pre-print
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,
more » ... 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.
arXiv:2207.00140v2 fatcat:c42dlf2mlneenad5xwb4mlka54