Counting Number Fields by Discriminant

The central topic of this dissertation is counting number fields ordered by discriminant. We fix a base field k and let Nd(k,G;X) be the number of extensions N/k up to isomorphism with Nk/Q(dN/k) ≤ X, [N : k] = d and the Galois closure of N/k is equal to G. We establish two main results in this work. In the first result we establish upper bounds for N|G| (k,G;X) in the case that G is a finite group with an abelian normal subgroup. Further, we establish upper bounds for the case N |F| (k,G;X) where G is a Frobenius group with an abelian Frobenius kernel F. In the second result we establish is an asymptotic expression for N6(Q;A4;X). We show that N6(Q,A4;X) = CX1/2 + O(X0.426...) and indicate what is expecedted under the `-torsion conjecture and the Lindelöf Hypothesis. We begin this work by stating the results that are established here precisely, and giving a historical overview of the problem of counting number fields. In Chapter 2, we establish background material in the areas of ramification of prime numbers and analytic number theory. In Chapter 3, we establish the asymptotic result for N6(Q,A4;X). In Chapter 4, we establish upper bounds for Nd(k,G;X) for groups with a normal abelian subgroup and for Frobenius groups. Finally we conclude with Chapter 5 with certain extensions of the method. In particular, we indicate how to count extensions of different degrees and discuss how to use tools about average results on the size of the torsion of the class group on almost all extensions in a certain family