Antoine Klughertz

The inverse Galois problem (IGP) asks the natural question "Is every finite group the Galois group of a finite Galois extension over the rationals ?". The question is still open but we know the answer is positive for large families of groups (e.g. abelian, symmetric, solvable..). At the beginning of the 20th century, Noether proposed a strategy to solve IGP based on Hilbert's proof of Sn being a Galois group over \Q. The talk will begin with a novice-friendly introduction to Galois theory of field extensions in order to state the inverse Galois problem and solve it in easy cases (abelian groups). I will then introduce Noether's strategy to answer IGP by using Hilbert's irreducibility theorem and explain to what extent it actually solves IGP. If time permits I may add elliptic curves related stuff or talk about weak IGP.