Real fields with small Galois groups

Number fields with solvable Galois groups and small Galois root discriminants

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 possib...

Galois Groups over Nonrigid Fields

Let F be a field with charF 6= 2. We show that there are two groups of order 32, respectively 64, such that a field F with char F 6= 2 is nonrigid if and only if at least one of the two groups is realizable as a Galois group over F . The realizability of those groups turns out to be equivalent to the realizability of certain quotients (of order 16, respectively 32). Using known results on conne...

Minimal Discriminants for Fields with Small Frobenius Groups as Galois Groups

Computation of Galois groups over function fields

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1, t2, . . . , tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.

Densities of Quartic Fields with Even Galois Groups

Let N(d,G,X) be the number of degree d number fields K with Galois group G and whose discriminant DK satisfies |DK | ≤ X. Under standard conjectures in diophantine geometry, we show that N(4, A4, X) X2/3+ , and that there are N3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S4, confirming a question of Gallagher. Unconditionally we hav...