Carlo Sircana

• The construction of number fields with given Galois group fits into the framework of the inverse Galois problem. This problem remains still unsolved, although many partial results have been obtained over the last century. Shafarevich proved in 1954 that every solvable group is realizable as the Galois group of a number field. Unfortunately, the proof does not provide a method to explicitly find such a field. This work aims at producing a constructive version of the theorem by solving the following task: given a solvable group $G$ and a $B\in \mathbf N$, construct all normal number fields with Galois group $G$ and absolute discriminant bounded by $B$. Since a field with solvable Galois group can be realized as a tower of abelian extensions, the main role in our algorithm is played by class field theory, which is the subject of the first part of this work. The second half is devoted to the study of the relation between the group structure and the field through Galois correspondence. In particular, we study the existence of obstructions to embedding problems and some criteria to predict the Galois group of an extension.