On polynomials over finite fields that are free of binomials

Abstract

Let \(\mathbb {F}_q\) be the finite field with q elements, where q is a power of a prime p. Given a monic polynomial \(f \in \mathbb {F}_q[x]\) that is not divisible by x, there exists a positive integer \(e=e(f)\) such that f(x) divides the binomial \(x^e-1\) and e is minimal with this property. The integer e is commonly known as the order of f and we write \(\textrm{ord}(f)=e\). Motivated by a recent work of the second author on primitive k-normal elements over finite fields, in this paper we introduce the concept of polynomials free of binomials. These are the polynomials \(f \in \mathbb {F}_q[x]\), not divisible by x, such that f(x) does not divide any binomial \(x^d-\delta \in \mathbb {F}_q[x]\) with \(1\le d<\textrm{ord}(f)\). We obtain some general results on polynomials free of binomials and we focus on the problem of describing the set of degrees of the polynomials that are free of binomials and whose order is fixed. In particular, we completely describe such set when the order equals a positive integer \(n>1\) whose prime factors divide \(p(q-1)\). Moreover, we also provide a correspondence between the polynomials that are free of binomials and cyclic codes that cannot be submerged into smaller constacyclic codes.