Construction of irreducible polynomials through rational transformations

Let ${\mathbb{F}}_q$ be the finite field with $q$ elements, where $q$ is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over ${\mathbb{F}}_q$ of high degree using rational transformations. In particular, given a divisor $D>2$ of $q+1$ and an irreducible polynomial $f\in {\mathbb{F}}_q[x]$ of degree $n$ such that $n$ is even or $D\not \equiv 2\pmod 4$,weshowhowtoobtainfrom$f$asequence${f_i}_{i\ge 0}$ of irreducible polynomials over ${\mathbb{F}}_q$ with $$\mathrm{d}\mathrm{e}\mathrm{g}$(f_i)=n\cdot D^{i}$.