. On the conjugate lattices, the modular group acts as . It follows that has Galois group isomorphic to over . Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing to polynomials with Galois group over The groups include infinitely many non-solvable groups. Notes References Alexander M. Macbeath, Extensions of the Rationals with Galois Group PGL(2,Zn), Bull. London Math. Soc., 1 (1969),332-338. Helmut Völklein, Groups as Galois Groups, an Introduction, Cambridge University Press, 1996. ISBN 978-0521065030 . Gunter Malle, Heinrich Matzat, Inverse Galois Theory, Springer-Verlag, 1999, . Gunter Malle, Heinrich Matzat, Inverse Galois Theory, 2nd edition, Springer-Verlag, 2018. Alexander Schmidt, Kay Wingberg, Safarevic's Theorem on Solvable Groups as Galois Groups (see also ) Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic Polynomials, Constructive Aspects of the Inverse Galois Problem, Cambridge University Press, 2002. Galois theory Unsolved problems in mathematics