On Mutual Definability of Operations on Fields

R. M. Korotkova, O. V. Kudinov, A. S. Morozov

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1032-1039
Number of pages8
JournalSiberian Mathematical Journal
Issue number6
Publication statusPublished - 1 Nov 2019


  • definability
  • exponentiation
  • field


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