A note in inverse Galois theory. Abstract:
The note explores connections between inverse Galois theory and Hilbert irreducibility, presenting results in the form of theorems and lemmas. The main focus is on establishing conditions under which a finite group can be realized as a Galois group over the rational numbers. The note introduces a corresponding polynomial associated with a finite group and explores its irreducibility over specific varieties. The main results include Theorem 1, which establishes conditions for a group to be a Galois group, and Theorem 2, which demonstrates the equivalence between the applicability of Hilbert irreducibility to the corresponding polynomial and the realizability of every finite group as a Galois group over the rational numbers. The note concludes with corollaries and lemmas supporting the main theorems. It can be downloaded as pdf from here.