Products of absolute Galois groups

Abstract

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 absolute Galois groups are known to be absolute Galois groups. The same is true about free pro-p products of absolute Galois groups which are pro-p groups. We show that, conversely, if C is a class of finite groups closed under forming subgroups, quotients, and extensions, and if the class of pro-C absolute Galois groups is closed under free pro-C products, then C is either the class of all finite groups or the class of all finite p-groups. As a tool, we prove a generalization of an old theorem of Neukirch which is of interest in its own right: if K is a non-Henselian field, then every finite group is a subquotient of GK, unless all decomposition subgroups of GK are pro-p groups for a fixed prime p.