On the Galois group of the modular equation

This thesis looks at a method of generating infinitely many extensions of the rationals with Galois group PGL(_2)Z(_n)). Firstly, the Galois group of the modular equation over Q(j) is shown to be PGL(_2)(Z(_n) by considering the n-th division points on an elliptic curve. Then, using Hilbert's Irreducibility Theorem and work discussed by Lang, we show that there are infinitely many rational values of such that this Galois group does not reduce in size. Finally, an equation whose roots generate the same extension as the modular equation but which has much smaller coefficients is found, based on work by Cohn.