Noether’s problem and the existence of generic polynomials

Abstract

The main topics of this thesis are Noether’s problem and the existence of generic polynomials. These problems are both related to the inverse Galois problem, which asks the question whether every finite group is isomorphic to the Galois group of a Galois extension over the field of rational numbers. We solved Noether’s problem and found generic polynomials for the subgroups of Sn for n ≤ 4 and the quaternion group of order 8. Moreover, we established generic polyomials for the cyclic groups of odd order and discussed their existence for some other groups such as the dihedral groups of odd order, p-groups and Frobenius groups. We also worked out a counterexample for Noether’s problem, namely the cyclic group of order 8.