The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace


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Cakiroglu Y., YAYLA O., Yilmaz E. S.

DESIGNS CODES AND CRYPTOGRAPHY, vol.90, no.10, pp.2407-2417, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 90 Issue: 10
  • Publication Date: 2022
  • Doi Number: 10.1007/s10623-022-01088-2
  • Journal Name: DESIGNS CODES AND CRYPTOGRAPHY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, PASCAL, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH
  • Page Numbers: pp.2407-2417
  • Keywords: Irreducible polynomials, Finite fields, Trace function, Algebraic curves, CURVES, ENUMERATION, GF(2)
  • Hacettepe University Affiliated: Yes

Abstract

We present the formula for the number of monic irreducible polynomials of degree n over the finite field F-q where the coefficients of x(n)(-1) and x vanish for n >= 3. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements a is an element of F-qn for which Trace(a) = 0 and Trace(a(-1)) = 0.