Editing Inverse Galois problem (section)

==A construction with an elliptic modular function==
Let {{math|''n'' > 1}} be any integer. A lattice {{math|Λ}} in the [[complex plane]] with period ratio {{mvar|τ}} has a sublattice {{math|Λ′}} with period ratio {{math|''nτ''}}. The latter lattice is one of a finite set of sublattices permuted by the [[modular group]] {{math|PSL(2, '''Z''')}}, which is based on changes of basis for {{math|Λ}}. Let {{mvar|j}} denote the [[elliptic modular function]] of [[Felix Klein]]. Define the polynomial {{math|''φ<sub>n</sub>''}} as the product of the differences {{math|(''X'' − ''j''(Λ<sub>''i''</sub>))}} over the conjugate sublattices. As a polynomial in {{mvar|X}}, {{math|''φ<sub>n</sub>''}} has coefficients that are polynomials over <math>\mathbb{Q}</math> in {{math|''j''(''τ'')}}.

On the conjugate lattices, the modular group acts as {{math|PGL(2, '''Z'''/''n'''''Z''')}}. It follows that {{math|''φ<sub>n</sub>''}} has Galois group isomorphic to {{math|PGL(2, '''Z'''/''n'''''Z''')}} over <math>\mathbb{Q}(\mathrm{J}(\tau))</math>.

Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing {{math|''φ<sub>n</sub>''}} to polynomials with Galois group {{math|PGL(2, '''Z'''/''n'''''Z''')}} over {{nowrap|<math>\mathbb{Q}</math>.}} The groups {{math|PGL(2, '''Z'''/''n'''''Z''')}} include infinitely many non-solvable groups.