Given a finite field F_q, constructing irreducible polynomials of given (usually high) degree n is an important questions in the arithmetic of finite fields (it is for instance crucial for the construction of the field extension field F_{q^n}). Depending on the application in mind, we usually require this irreducible polynomial to have further interesting properties (being self-reciprocal, normal etc). In the past, various iterative recursive constructions of such polynomials have been proposed. Similarly, recently, questions about iterative compositions of polynomials, about the irreducibility of the resulting polynomials and their possible factorizations, related questions about arboreal Galois representations and the corresponding dynamical systems have been studied extensively. In both cases many of the results can be interpreted and better understood by considering particular extensions of algebraic function fields, their Galois closures, corresponding Galois groups and the splitting of primes. Interesting phenomena can usually be explained using simple group theoretical results and can be readily generalized. Many interesting examples known in the literature are better understood by interpreting them using interesting subfields of rational function fields (cyclic Kummer or Artin-Schreier extensions corresponding to cyclic subgroups of the projective linear group). We will try to better understand these constructions, consider various other non-cyclic subgroups of the projective linear group and similar constructions using extensions involving elliptic or hyperelliptic curves.
