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

The paper is devoted to constructive theory of synthesis of irreducible polynomials and irreducible N-polynomials (with linearly independent roots) over finite fields. For a suitably chosen initial N-polynomial F1(x) F2s[x] of degree n, polynomials Fk(x) F2s[x] of degrees 2k−1n are constructed by iteration of the transformation of variable x→x+δ2x−1, where δ F2s and δ≠0. It is shown that the set of roots of the polynomials Fk(x) forms a normal basis of F22k−1sn over F2s. In addition, the sequences are trace-compatible in the sense that the trace relation maps the corresponding roots onto each other. Furthermore, for a prime power q=ps, some recurrent methods for constructing families of monic irreducible polynomials of degree npk, k 1, over Fq is given. This construction is a generalization of Varshamovʹs construction given for prime fields. The construction gives an iterative technique to construct sequences (Fk(x)k 0) of N-polynomials of degree pk+2 over Fq.