Factorization of the Degree of Sphenic Polynomials Over the Galois Fields of Arbitrary Characteristics

Abstract

By sphenic polynomials, we mean polynomials formed by the product of three (not necessarily different) irreducible polynomials with a priori unknown degree. The study's main goal is to develop an effective algorithm for factorizing degrees of sphenic polynomials with minimal computational complexity. Different solutions to the problem of factorization degrees of sphenic polynomials depending on the ratio degree of the cycle period of these polynomials consider. The sphenic polynomial cycle period defines as a parameter equal to the number of non-repeating subtractions computed on the linear-logarithmic scale of the group formed by the sphenic polynomial. The proposed algorithm is invariant to the characteristics of the Galois fields generated by the multipliers of sphenic polynomials. Numerous numerical examples confirm the correctness of the results. Directions for further research outlines.