a transposition in is implied. Any finite doubly transitive permutation group containing a transposition is a full symmetric group. Hilbert's irreducibility theorem then implies that an infinite set of rational numbers give specializations of whose Galois groups are over the rational field In fact this set of rational numbers is dense in The discriminant of equals and this is not in general a perfect square. Alternating groups Solutions for alternating groups must be handled differently for odd and even degrees. Odd Degree Let Under this substitution the discriminant of equals which is a perfect square when is odd. Even Degree Let: Under this substitution the discriminant of equals: which is a perfect square when is even. Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups. Rigid groups Suppose that are conjugacy classes of a finite group , and be the set of -tuples of such that is in and the product is trivial. Then is called rigid if it is nonempty, acts transitively on it by conjugation, and each element of generates . showed that if a finite group has a rigid set then it can often be realized as a Galois group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible characters of on the conjugacy classes .) This can be used to show that many finite simple groups, including the monster group, are Galois groups of extensions of the rationals. The monster group is generated by a triad of elements of orders , , and . All such triads are conjugate. The prototype for rigidity is the symmetric group , which is generated by an -cycle and a transposition whose product is an -cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group. A construction with an elliptic modular function Let be any integer. A lattice in the complex plane with period ratio has a sublattice with period ratio . The latter lattice is one of a finite set of sublattices permuted by the modular group , which is based on changes of basis for . Let denote the elliptic modular function of Felix Klein. Define the polynomial as the product of the differences over the conjugate sublattices. As a polynomial in , has coefficients that are polynomials over in