The Galois Inverse Problem Over The Field Of Rational Numbers

This paper is a review report on the research status of the inverse Galois problem in the field of rational numbers,that is,whether a given finite group is a Galois group that extends a finite Galois group in the field of rational numbers.In this paper,we first introduce some concrete finite groups,such as symmetry groups and finite Abelian groups,and then we study the general finite groups by using the language of algebraic geometry and algebraic number theory.In this paper,we will prove that the Gallois inverse problem over the field of rational numbers holds for any given finite group,assuming that the Colliot-Thelene conjecture holds.