Irreducible polynomials from a cubic transformation

On a type of permutation rational functions over finite fields

“…Equivalence of rational functions over finite fields also arises in other circumstances. There is a construction of irreducible polynomials over F q using a rational function R(X) ∈ F q [X]; the number of irreducible polynomials produced by the construction depends only on the equivalence class of R(X) [15]. It is known that the equivalence classes of rational functions f ∈ F q (X) \ F q such that F q (X)/F q (f ) is Galois are in one-to-one correspondence with the classes of conjugate subgroups of PGL(2, F q ); see [12].…”