Invariant rational functions, linear fractional transformations and irreducible polynomials over finite fields

For a subgroup of PGL(2,q) we show how some irreducible polynomials over Fq arise from the field invariant rational functions. The proofs rely on combining two actions PGL(2,F), one projective line F and other function F(x). functions in F(x) are used to that regular patterns exist factorization certain into polynomials. We use results about group orbit polynomial, whose included.