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Sibirskii Matematicheskii Zhurnal, 2019, Volume 60, Number 6, Pages 1324–1334
DOI: https://doi.org/10.33048/smzh.2019.60.611 (Mi smj3152) 		 
On mutual definability of operations on fields

R. M. Korotkovaa, O. V. Kudinovba, A. S. Morozovba

a Novosibirsk State University, Mechanics and Mathematics Department
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
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DOI: https://doi.org/10.33048/smzh.2019.60.611
Abstract: We study the possibilities of defining some operations on fields via the remaining operations. In particular, we prove that multiplication on an arbitrary field can be defined via addition if and only if the field is a finite extension of its prime subfield. We give a sufficient condition for the nondefinability of addition via multiplication and demonstrate that multiplication and addition on the reals and complexes cannot be mutually defined by means of the relations with parameters which are preserved under automorphisms. We also describe the mutual definability of addition, multiplication, and exponentiation via the remaining two operations.
Funding agency Grant number
Siberian Branch of Russian Academy of Sciences I.1.1 (проект № 0314–2019–0001)
1.1.1 (проект № 0314–2019–0003)
Russian Foundation for Basic Research 17-01-00247_а
O. V. Kudinov was supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant No. I.1.1, Project 0314—2019—0001). A. S. Morozov was supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant No. I.1.1, Project 0314—2019—0003). O. V. Kudinov and A. S. Morozov were partially supported by the Russian Foundation for Basic Research (Grant 17—01—00247).
Received: 13.07.2018
Revised: 08.07.2019
Accepted: 24.07.2019
English version:
Siberian Mathematical Journal, 2019, Volume 60, Issue 6, Pages 1032–1039
DOI: https://doi.org/10.1134/S0037446619060119
Bibliographic databases:
Document Type: Article
UDC: 510.67
Language: Russian
Citation: R. M. Korotkova, O. V. Kudinov, A. S. Morozov, “On mutual definability of operations on fields”, Sibirsk. Mat. Zh., 60:6 (2019), 1324–1334; Siberian Math. J., 60:6 (2019), 1032–1039
Citation in format AMSBIB
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\paper On mutual definability of operations on fields
\jour Sibirsk. Mat. Zh.
\yr 2019
\vol 60
\issue 6
\pages 1324--1334
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\crossref{https://doi.org/10.33048/smzh.2019.60.611}
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\jour Siberian Math. J.
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\issue 6
\pages 1032--1039
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