  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 $D\not \equiv
2\pmod 4$, we show how to obtain from $f$ a sequence $\{f_i\}_{i\ge 0}$ of
irreducible polynomials over $\mathbb F_q$ with $\mathrm{deg}(f_i)=n\cdot
D^{i}$.
