RRC-fields with small absolute Galois groups

Products of absolute Galois groups ∗

If the absolute Galois group GK of a field K is a direct product GK = G1 × G2 then one of the factors is prosolvable and either G1 and G2 have coprime order or K is henselian and the direct product decomposition reflects the ramification structure of GK . So, typically, the direct product of two absolute Galois groups is not an absolute Galois group. In contrast, free (profinite) products of ab...

Relatively projective groups as absolute Galois groups

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

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...

Free Products of Absolute Galois Groups *