A number theoretical approach to polynomials over finite fields

Let q be a prime power and Fq be the finite field with q elements. The explicit constructions of irreducible polynomials over Fq of high degree is one of the main problems in the arithmetic of finite fields which has many applications in several areas such as coding theory and cryptography. In general, some recursive methods are preferred to do these constructions using rational transformations. In particular, we are interested in methods that are obtained by using quadratic transformations. For doing this, we will first classify and normalize the rational transformations of degree 2 using the behaviour of the ramified places in the corresponding rational function field extensions over the finite field Fq. Then we will investigate the constructions using Galois theory and some basic observations in group theory. This approach helps to better understand the iterative constructions and gives various generalizations of them. It also enables to determine the requirements put on the initial polynomials.