Springer, Caleb;
(2023)
Definability and decidability for rings of integers in totally imaginary fields.
Bulletin of the London Mathematical Society
10.1112/blms.12933.
(In press).
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Abstract
We show that the ring of integers of is existentially definable in the ring of integers of , where denotes the field of all totally real numbers. This implies that the ring of integers of is undecidable and first‐order nondefinable in . More generally, when is a totally imaginary quadratic extension of a totally real field , we use the unit groups of orders to produce existentially definable totally real subsets . Under certain conditions on , including the so‐called ‐number of being the minimal value , we deduce the undecidability of . This extends previous work that proved an analogous result in the opposite case . In particular, unlike prior work, we do not require that contains only finitely many roots of unity.
| Type: | Article |
|---|---|
| Title: | Definability and decidability for rings of integers in totally imaginary fields |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1112/blms.12933 |
| Publisher version: | https://doi.org/10.1112/blms.12933 |
| Language: | English |
| Additional information: | © 2023 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10178577 |
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