A Note on Powers in Finite Fields

Andreas Aabrandt, Vagn Lundsgaard Hansen

Research output: Contribution to journalJournal articleResearchpeer-review

499 Downloads (Pure)


The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation which is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.
Original languageEnglish
JournalInternational Journal of Mathematical Education in Science and Technology
Issue number6
Pages (from-to)987-991
Publication statusPublished - 2016


  • Finite fields
  • Prime numbers
  • Squares and powers in finite fields

Cite this