On Mutual Definability of Operations on Fields

On Mutual Definability of Operations on Fields

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 no...

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Bibliographic Details
Published in:Siberian mathematical journal Vol. 60; no. 6; pp. 1032 - 1039
Main Authors: Korotkova, R. M., Kudinov, O. V., Morozov, A. S.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01-11-2019
Springer Nature B.V
Subjects:
Automorphisms
Mathematics
Mathematics and Statistics
Multiplication
field
exponentiation
definability
Online Access:Get full text
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Description
Summary: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.
ISSN:0037-4466
1573-9260
DOI:10.1134/S0037446619060119