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Let $S$ be a rational fraction and let $f$ be a polynomial over a finitefield. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certaincases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. Forinstance, in odd characteristic, this is the case for the rational fraction$S=(x^2+1)/(2x)$, known as the $R$-transform, and for a positive density of allirreducible polynomials $f$.  We interpret these transforms in terms of isogenies of elliptic curves. Usingcomplex multiplication theory, we devise algorithms to generate a large numberof other rational fractions $S$, each of which yields infinite families ofirreducible polynomials for a positive density of starting irreduciblepolynomials $f$.

Iterative constructions of irreducible polynomials from isogenies