Galois groups of prime degree and the O'Nan-Scott theorem

Discussion begins with a modular method for determining cycle types of permutations in the Galois group of a given separable irreducible polynomial over Q. As the Galois group is a transitive permutation group on the n roots of its irreducible polynomial, a list of all transitive groups of degree n, together with the cycle type distributions of each group, allows a probablistic determination of the group in a process of elimination. In the case of prime degree, transitive groups are primitive and by the O'Nan-Scott theorem are of restricted form. Theory is presented in order to arrive at these results and others, so that in conjunction with the classification theorem on finite simple groups, it is possible (in principle) to list all primitive permutation groups of particular prime degree. The case of degree 17 is examined to obtain a list of the transitive permutation groups of degree seventeen, as well as the cycle type of distributions of the groups identified.