Submission date: 4 March 2024

Abstract:

Let TFAb_r be the class of torsion-free abelian groups of rank r, and let FD_r be the class of fields of characteristic 0 and transcendence degree r. We compare these classes using various notions. Considering Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the TFAb_r are strictly increasing under Borel reducibility. This is not so for the classes FD_r. Thomas and Velickovic showed that for sufficiently large r, the classes FD_r are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of TFAb_r in FD_r, and of FD_r in FD_{r+1}. We show that under computable countable reducibility, TFAb_1 lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on TFAb_1 lies on top among equivalence relations that are effective Σ_3, along with the Vitali equivalence relation on 2^\omega.

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 2403.02488: pdf, ps.