Construction of irreducible polynomials through rational transformations

Abstract

Let Fq be the finite field with q elements, where q is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over Fq of high degree using rational transformations. In particular, given a divisor D>2 of q+1 and an irreducible polynomial f∈Fq[x] of degree n such that n is even or D≢2(mod4), we show how to obtain from f a sequence {fi}i≥0 of irreducible polynomials over Fq with deg⁡(fi)=n⋅Di.