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TORSION-FREE ABELIAN GROUPS OF FINITE RANK AND FIELDS OF FINITE TRANSCENDENCE DEGREE

Part of: Computability and recursion theory Model theory

Published online by Cambridge University Press:  13 February 2025

MENG-CHE TURBO HO Open the ORCID record for MENG-CHE TURBO HO [Opens in a new window] ,
JULIA FRANDSEN KNIGHT  and
RUSSELL GEDDES MILLER Open the ORCID record for RUSSELL GEDDES MILLER [Opens in a new window]
MENG-CHE TURBO HO*
Affiliation:
DEPARTMENT OF MATHEMATICS CALIFORNIA STATE UNIVERSITY, NORTHRIDGE 18111 NORDHOFF STREET, NORTHRIDGE CA 91330, USA
JULIA FRANDSEN KNIGHT
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME 255 HURLEY HALL, NOTRE DAME IN 46556-5641, USA E-mail: julia.f.knight.1@nd.edu
RUSSELL GEDDES MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS QUEENS COLLEGE 65-30 KISSENA BLVD. QUEENS, NY 11367 USA AND PH.D. PROGRAMS IN MATHEMATICS AND COMPUTER SCIENCE CUNY GRADUATE CENTER 365 FIFTH AVENUE NEW YORK, NY 10016 USA E-mail: russell.miller@qc.cuny.edu
*
E-mail: turboho@gmail.com
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Abstract

Let $\operatorname {TFAb}_r$ be the class of torsion-free abelian groups of rank r, and let $\operatorname {FD}_r$ be the class of fields of characteristic $0$ and transcendence degree r. We compare these classes using various notions. Considering the 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 $\operatorname {TFAb}_r$ are strictly increasing under Borel reducibility. This is not so for the classes $\operatorname {FD}_r$. Thomas and Velickovic showed that for sufficiently large r, the classes $\operatorname {FD}_r$ 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 $\operatorname {TFAb}_r$ in $\operatorname {FD}_r$, and of $\operatorname {FD}_r$ in $\operatorname {FD}_{r+1}$. We show that under computable countable reducibility, $\operatorname {TFAb}_1$ lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on $\operatorname {TFAb}_1$ lies on top among equivalence relations that are effective $\Sigma _3$.

Keywords

isomorphism problemtorsion-free abelian groupsfields with finite transcendence degree

MSC classification

Primary: 03C57: Effective and recursion-theoretic model theory 03D45: Theory of numerations, effectively presented structures 03D55: Hierarchies

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 30
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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