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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,arXiv:2207.00140v2 fatcat:c42dlf2mlneenad5xwb4mlka54