Abstract
We consider fields computable in polynomial time (P-computable). We prove that under some assumptions about a P-computable field \( (A, +, \cdot ) \) of characteristic 0, there exists a P-computable field \( (B, +, \cdot ) \cong (A, +, \cdot ) \), in which \(x^{-1}\) is not a primitive recursive function. In particular, this holds for the field \( \mathbb Q \) of rational numbers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Cenzer, D., Remmel, J.: Polynomial time versus recursive models. Ann. Pure Appl. Logic 54(1), 17–58 (1991)
Alaev, P.: Inversion operations in algebraic structures. Computability 12(4), 315–322 (2023)
Alaev, P.E.: Complexity of the inversion operations in groups. Algebra Log. 62(2), 103–118 (2023)
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)
Van der Waerden B.L.: Algebra I. Springer, Heidelberg (1971)
Acknowledgments
This study was supported by the Russian Science Foundation (grant No. 23-11-00170), https://rscf.ru/project/23-11-00170.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Ethics declarations
Disclosure of Interests
The author has no competing interests to declare that are relevant to the content of this article.
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Alaev, P. (2024). Inversion in P-Computable Fields. In: Levy Patey, L., Pimentel, E., Galeotti, L., Manea, F. (eds) Twenty Years of Theoretical and Practical Synergies. CiE 2024. Lecture Notes in Computer Science, vol 14773. Springer, Cham. https://doi.org/10.1007/978-3-031-64309-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-64309-5_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-64308-8
Online ISBN: 978-3-031-64309-5
eBook Packages: Computer ScienceComputer Science (R0)