Inverse galois problem

Abstract

The main focus of this thesis is so called Inverse Galois Problem. The statement of the problem is that given a finite group G, does there exist a finite Galois extension L/Q whose Galois group is G? There has been a great process in the problem, but it is still open. Galois theory is the study of the structure and symmetry of a polynomials or associated field extensions. According to the Fundamental Theorem of Galois Theory, there exists a correspondence between a finite algebraic field extension and its Galois group. But, this correspondence is very complicated in general. Inverse Galois Problem deals with this complexity. We will give an introduction to the Inverse Galois Problem and present some different approaches to construct an extension of Q that gives a desired Galois group. In particular, we will realize some specific groups as Galois groups, these groups are finite abelian groups, symmetric groups Sn, the general linear group GL2(Fp), and the projective special linear group PSL2(Fp). Finally, we will give a short survey about known results on Inverse Galois Problem.