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

We give iterative constructions for irreducible polynomials over F(q )of degree n & sdot;t (R) for all r >= 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 >= 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.