On Mutual Definability of Operations on Fields
- Authors: Korotkova R.M.1, Kudinov O.V.2, Morozov A.S.2
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Affiliations:
- Novosibirsk State University
- Sobolev Institute of Mathematics
- Issue: Vol 60, No 6 (2019)
- Pages: 1032-1039
- Section: Article
- URL: https://journals.rcsi.science/0037-4466/article/view/172745
- DOI: https://doi.org/10.1134/S0037446619060119
- ID: 172745
Cite item
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.
Keywords
About the authors
R. M. Korotkova
Novosibirsk State University
Author for correspondence.
Email: rozulka93@mail.ru
Russian Federation, Novosibirsk
O. V. Kudinov
Sobolev Institute of Mathematics
Author for correspondence.
Email: kud@math.nsc.ru
Russian Federation, Novosibirsk
A. S. Morozov
Sobolev Institute of Mathematics
Author for correspondence.
Email: morozov@math.nsc.ru
Russian Federation, Novosibirsk
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