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On mutual definability of operations on fields. (English. Russian original) Zbl 1468.03044

Sib. Math. J. 60, No. 6, 1032-1039 (2019); translation from Sib. Mat. Zh. 60, No. 6, 1324-1334 (2019).
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.

MSC:

03C57 Computable structure theory, computable model theory
03C40 Interpolation, preservation, definability
03D45 Theory of numerations, effectively presented structures
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