r-primitive k-normal polynomials over finite fields with last two coefficients prescribed

2025/01/09 by Chatterjee, K., Sharma, R. K., Tiwari, S. K.
#11T23 #12E20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.04999

Abstract

Let ξ∈\mathbbF_q^m be an r-primitive k-normal element over \mathbbF_q, where q is a prime power and m is a positive integer. The minimal polynomial of ξ is referred to be the r-primitive k-normal polynomial of ξ over \mathbbF_q. In this article, we study the existence of an r-primitive k-normal polynomial over \mathbbF_q such that the last two coefficients are prescribed. In this context, first, we prove a sufficient condition which guarantees the existence of such a polynomial. Further, we compute all possible exceptional pairs (q,m) in case of 3-primitive 1-normal polynomials for m≥ 7.

Related

• On r-primitive k-normal polynomials with two prescribed coefficients
• About r- primitive and k-normal elements in finite fields
• On the Coefficients of Primitive Normal polynomials
• Pairs of r-primitive and k-normal elements in finite fields
• Inverses of r-primitive k-normal elements over finite fields