Iterated constructions of irreducible polynomials over finite fields with linearly independent roots (Q596584)

The paper presents constructions of sequences \((F_k(x))_{k\geq1}\) of normal polynomials of degree \(np^{k-1}\) over a finite field of characteristic \(p\), in such a way that starting from a suitable initial polynomial the polynomials \(F_k(x)\) are iteratively generated by a simple rational transformation. In case \(p=2\), the iterated transformation \(x\mapsto x+\delta^2x^{-1}\) with \(\delta\in\mathbb{F}_{2^s}^*\) produces such a sequence from a suitably chosen polynomial \(F_1(x)\in\mathbb{F}_{2^s}[x]\). (In constrast to comparable constructions, the degree of \(F_1(x)\) is arbitrary here.) Furthermore, some recursive methods for constructing families of irreducible polynomials of degree \(np^{k-1}\) over a finite characteristic \(p\) field are given. This technique is a generalization of a construction given by Varshamov for prime fields. In addition, it yields an iterative method to generate sequences of normal polynomials of degree \(p^k\) (\(k=2,3,\ldots\)) over a finite field of characteristic \(p\).