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
Famous quotes containing the words construction and/or function:
“When the leaders choose to make themselves bidders at an auction of popularity, their talents, in the construction of the state, will be of no service. They will become flatterers instead of legislators; the instruments, not the guides, of the people.”
—Edmund Burke (1729–1797)
“Advocating the mere tolerance of difference between women is the grossest reformism. It is a total denial of the creative function of difference in our lives. Difference must be not merely tolerated, but seen as a fund of necessary polarities between which our creativity can spark like a dialectic.”
—Audre Lorde (1934–1992)