Inverse Galois problem

In this thesis, we presente and investigate the inverse Galois problem: for a given finite group, is it possible to find a Galois extension whose Galois group is the given group? We begin by describing the situations in which the problem always has a solution. We then present the main methods used to find a solution for specific types of groups and fields, focusing all attention on the case of extensions of Q (the rational field). The first method considered is based on the so called embedding problem: we follow all steps necessary to survey the Kronecker-Weber theorem for abelian groups, the Scholz-Reichardt theorem for p-groups, and the main result of Shafarevich for solvable groups. We also apply this method to the case of the quaternion group Q_8. The second method we deal with is the method based on the rigidity criterion: this method has been developed for simple groups and solves the problem for most of them. It is based on identifying two key properties in the conjugacy classes of the group: rigidity and rationality. If these properties are satisfied, then the inverse Galois problem can be solved using a specific theorem based on rigidity criteria. In order to state and use the concepts of rigidity and rationality, we need to introduce some foundational results and techniques, such as the Riemann's Existence Theorem and Laurent series, which allow us to proceed toward solving the inverse Galois problem. Also in this case, the method is applied to the example of the simple groups PSL(2,p). The final method involves elliptic curves, and it is the most recent of the three. We show how elliptic curves can be used to solve the inverse Galois problem and then we apply this method to a significant example.