Construction of nonabelian etale fundamental groups

Abstract

In this article, under the assumption of the GRH(Generalized Riemann Hypothesis), we show that there are real quadratic fields K such that the etale fundamental group of the spectrum of the ring of integers is isomorphic to A_5. The proof uses standard methods involving Odlyzko bounds as well as the proof of Serre's modularity conjecture.We will also prove that for any finite solvable group G, there exist infinitely many abelian extensions K/Q and Galois extensions M/Q such that the Galois group Gal(M/K) is isomorphic to G and M/K is unramified. In general, the construction problem of unramified Galois extensions M/K of number fields with given Galois groups is not so difficult if we impose no restrictions (on the degree of K, on the maximality of M/K, etc.). However, if we require degree of K to be "small" for "big" given Galois group Gal(M/K), the problem turns to be highly difficult.We will apply theory of embedding problem of Galois extensions to this problem and gives a recursive procedure to construct such extensions so that whose base field has possibly small degree.