Number fields with solvable Galois groups and small Galois root discriminants

Abstract

We apply class field theory to compute complete tables of number fields with Galois root discriminant less than 8πeγ . This includes all solvable Galois groups which appear in degree less than 10, groups of order less than 24, and all dihedral groups Dp where p is prime. Many people have studied questions of constructing complete lists of number fields subject to conditions on degree and possibly Galois group, with a goal of determining complete lists of such fields with discriminant less than a fixed bound. This study can be phrased in those terms, with the principle distinction being that our discriminant bounds apply to Galois fields, rather than to particular stem fields. For a finite group G and bound B > 0, let K(G,B) be the set of number fields in C which are Galois over Q and which have root discriminant ≤ B. It is a classical theorem that each set K(G,B) is finite. The first author and Roberts [JR07a] considered these sets with B = Ω = 8πe ≈ 44.7632, a constant introduced by Serre [Ser86]. They conjecture that there are only finitely many such fields in the union of the K(G,Ω), with G running through all finite groups. They determine all sets K(A,Ω) where A is an abelian group, and most sets K(G,Ω) for groups G which appear as Galois groups of irreducible polynomials of degree at most 6. Here we extend those results using class field theory. We complete their study of Galois groups which appear in degree 6, all solvable extensions in degrees 7–9, as well as for all groups G with |G| ≤ 23. Finally, we determine the sets K(D`,Ω) where ` is an odd prime. The primary theoretical difficulty in carrying out these computations is to deduce bounds which effectively screen out conductors for our extensions. Section 1 sets notation and describes some of the methods used before Section 2 addresses this question, and describes our overall process. Section 3 summarizes the results of our computations. 1. Notation and Conventions 1.1. Groups. We let Cn denote the cyclic group of order n, and Dn the dihedral group of order 2n. Direct products are indicated by juxtaposition, so G1 ×G2 will be denoted by simply G1G2, and G1 × G1 will be written G1. Many groups will be semidirect products, and N : H will denote a semidirect product with normal subgroup N and complimentary subgroup H. We will write G1 : G2 : G3 to mean (G1 : G2) : G3. If H ≤ Sn, then the wreath product is denoted G oH ∼= G : H. An extension of G1 by G2 will be denoted by G1.G2. 2010 Mathematics Subject Classification. Primary 11R21; Secondary 11R37.