A Construction With An Elliptic Modular Function
Let n be any integer greater than 1. A lattice Λ in the complex plane with period ratio τ has a sublattice Λ' with period ratio nτ. The latter lattice is one of a finite set of sublattices permuted by the modular group PSL(2,Z), which is based on changes of basis for Λ. Let j denote the elliptic modular function of Klein. Define the polynomial φn as the product of the differences (X-j(Λi)) over the conjugate sublattices. As a polynomial in X, φn has coefficients that are polynomials over Q in j(τ).
On the conjugate lattices, the modular group acts as PGL(2,Zn). It follows that φn has Galois group isomorphic to PGL(2,Zn) over Q(J(τ)).
Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing φn to polynomials with Galois group PGL(2,Zn) over Q. The groups PGL(2,Zn) include infinitely many non-solvable groups.
Read more about this topic: Inverse Galois Problem
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