Definability and undecidability via the torsion subgroup of units

@misc{springer2026definability,
  title = {{Definability and undecidability via the torsion subgroup of units}},
  author = {Caleb Springer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27210},
  eprint = {2609.27210},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27210},
}

TY  - UNPB
TI  - Definability and undecidability via the torsion subgroup of units
AU  - Caleb Springer
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27210
UR  - https://arxiv.org/abs/2609.27210
AB  - In this paper, we prove that ℤ is first-order definable in the ring of integers ℤ^ab of the maximal abelian extension ℚ^ab of ℚ, which implies that the first-order theory of ℤ^ab is undecidable. More generally, writing i = √(−1) and ℚ^tr for the field of all totally real numbers, we prove new definability and undecidability results for rings of integers of subfields of ℚ^tr(i), focusing especially on fields which contain infinitely many roots of unity. The key ingredient for these results is that there is a parameter-free positive-existential formula which defines the roots of unity μ(𝒪_L) inside 𝒪_L for every field L⊆ ℚ^tr(i).
ER  -