The R-transform as power map and its generalizations to higher degree

Abstract

We give iterative constructions for irreducible polynomials over $F_q$ of degree $n\cdot t^r$ for all $r\ge 0$, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions $x^t$. The R-transform introduced by Cohen is recovered as a particular case corresponding to $x^2$, hence we obtain a generalization of Cohen's R-transform ($t=2$) to arbitrary degrees $t\ge 2$. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of $F_q$ we recover and generalize a recursive construction of Panario, Reis and Wang.