Irreducible compositions of polynomials over finite fields of even characteristic

This paper presents some results with the constructive theory of synthesis of irreducible polynomials over a Galois field with even characteristic. We prove a theorem that plays an important role for constructing irreducible polynomials. By this theorem two recurrent methods for constructing families of irreducible polynomials of degree $n2^{k}~(k=1,2,\ldots )$ over $\mathbb F _{2^{s}}$ are proposed. It is shown that in this special case, the sequences of irreducible polynomials are N-polynomial of degree $2^{k}$ .