# Interpreting a field in its Heisenberg group

@article{Alvir2021InterpretingAF, title={Interpreting a field in its Heisenberg group}, author={Rachael Alvir and Wesley Calvert and Grant Goodman and Valentina S. Harizanov and Julia A. Knight and Andrey S. Morozov and Russell G. Miller and Alexandra A. Soskova and Rose Weisshaar}, journal={The Journal of Symbolic Logic}, year={2021} }

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 non-commuting 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 Harrison-Trainor, Melnikov, R. Miller…

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