A Brief Survey on the Inverse Galois Problem

Abstract

Inverse Galois problem (IGP) states whether any finite group is realizable as a Galois group over the field K. It is the question of the structure and representation of the Galois group and also questions its epimorphic images. So, it is called an inverse Galois problem. For K=ℚ (the field of rational number), it is called a classical inverse Galois problem (CIGP). This paper reviews the positive answer to the classical inverse Galois problem (CIGP) for all finite abelian groups and some finite non-abelian solvable groups. We also discuss this problem (CIGP) for some finite non-solvable groups in this paper. This problem still remains to solve, but if we find the true value of the statement ‘All subgroups of order m of the symmetric group (S_m) for all m are realizable as Galois group over ℚ’ then its truth value gives the answer of CIGP. We check this statement for m=1,2,3,4 and 5 in this paper, where we get that this statement is true. If this statement is true, then CIGP has a positive answer. But if this statement is false then CIGP has a negative answer.