Interpreting a field in its Heisenberg group
Abstract
We improve on and generalize a 1960 result of Maltsev. For a field $F$, we denote by $H(F)$ the Heisenberg group with entries in $F$. Maltsev showed that there is a copy of $F$ defined in $H(F)$, using existential formulas with an arbitrary noncommuting pair $(u,v)$ as parameters. We show that $F$ is interpreted in $H(F)$ using computable $\Sigma_1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of HarrisonTrainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of $F$ are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of $F$ represented by triples in $H(F)$. Looking at what was used to arrive at this parameterfree interpretation of $F$ in $H(F)$, we give general conditions sufficient to eliminate parameters from interpretations.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.11805
 Bibcode:
 2020arXiv200611805A
 Keywords:

 Mathematics  Logic;
 03C57 (Primary) 03D45;
 20H20;
 12L12
 EPrint:
 Published online by the *Journal of Symbolic Logic*, 23 December 2021. Print version to appear subsequently