Let \(F\) be a field containing a primitive \(p\)th root of unity. Let \(G_F\) denote the absolute Galois group of \(F\) and let \(q=p^d\) be a prime power. The descending \(q\)-central sequence of \(G\) is defined inductively by \[ G^{(1)}=G,\quad G^{(i+1)}=(G^{(i)})^q[G^{(i)},G],\quad i=1,2,\dots \] As usual, \([h,g]=h^{-1}g^{-1}hg\) denotes the commutator of \(h,g\).
The main theorem of the paper under review states the following: For \(p\neq 2\), \(G_F^{(3)}\) is the intersection of all open normal subgroups \(N\) of \(G_F\) such that \(G_F/N\) is isomorphic to one of \(1\), \({\mathbb{Z}}/p^2\), and \(M_{p^3}\).
Here \(M_{p^3}\) is the unique nonabelian group of order \(p^3\) and exponent \(p^2\). A similar result for \(p=2\) is already known. The proof of the main theorem is purely cohomological and gives a new proof for \(p=2\).