A new direction on constructing irreducible polynomials over finite fields
Kaimin Cheng · Finite Fields and Their Applications · 2024
Let q be a power of a prime p and F q be the finite field of order q . Let ϕ ( x ) be any polynomial in F q [ x ] and Φ ( x ) : = 1 ϕ ( x ) . For any positive integer n , denote Φ ( n ) to be the n -th iterate of Φ and d n , ϕ to be the denominator of Φ ( n ) . We call ϕ ( x ) ∈ F q [ x ] inversely stable over F q if d n , ϕ are distinct and irreducible over F q for all n . In this paper, we aim to find a class of inversely stable polynomials over F q . Actually, let ϕ ( x ) : = x p + a x + b ∈ F p [ x ] , it is proved that ϕ ( x ) is inversely stable over F p if and only if a = − 1 and b ≠ 0 ; moreover, if ϕ ( x ) is inversely stable over F p , then d n , ϕ is of degree p n for any positive integer n . Consequently, an infinite family of irreducible polynomials over F p is obtained.