Number fields with solvable Galois groups and small Galois root discriminants

Abstract

Abstract. We apply class field theory to compute complete tables of number fields with Galois root discriminant less than 8πe γ . This includes all solvable Galois groups which appear in degree less than 10, groups of order less than 24, and all dihedral groups D p 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. Jones and Roberts [JR07a] considered these sets with B = Ω = 8πe γ ≈ 44.7632, a constant introduced by Serre [Ser86]. This number is the asymptotic limit for root discriminant bounds assuming the Generalized Riemann Hypothesis. Jones and Roberts 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, treat 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.

A database of number fields

“…In a few cases, we obtained polynomials from other sources, notably for number fields of small discriminant: those compiled by the Bordeaux group [7], which in turn were computed by several authors, and the tables of totally real fields of Voight [39,40]. In addition, we include fields found by the authors in joint work with others [13,24].…”

Minimal solvable nonic fields

“…In [10], we address (with Wallington) the problem: given a finite group G and a bound B grd , to find all number fields K such that Gal(K gal /Q) ∼ = G and grd(K) B grd . In some ways, this grd-analog of our problem is simpler: by inequality (4.2) one has a fixed bound for all intermediate fields K j in the calculation, namely grd(K j ) B grd .…”

“…In some ways, this grd-analog of our problem is simpler: by inequality (4.2) one has a fixed bound for all intermediate fields K j in the calculation, namely grd(K j ) B grd . One has to deal with Galois root discriminants, but [10] provides means for handling this. In [10], the grd-analog of our problem is solved for solvable nonic Galois groups with bound B grd = 8πe γ ≈ 44.7632, a constant introduced by Serre [14].…”

“…One has to deal with Galois root discriminants, but [10] provides means for handling this. In [10], the grd-analog of our problem is solved for solvable nonic Galois groups with bound B grd = 8πe γ ≈ 44.7632, a constant introduced by Serre [14]. The origins of this constant will not play a role here, but it is the asymptotic limit for root discriminant bounds assuming the generalized Riemann hypothesis.…”

“…The origins of this constant will not play a role here, but it is the asymptotic limit for root discriminant bounds assuming the generalized Riemann hypothesis. In some cases, we extended the computations from [10] here.…”