Galois groups and cohomological functors

Let q = p^s be a prime power, F a field containing a root of unity of order q, and GF its absolute Galois group. We determine a new canonical quotient Gal(F₍₃₎/F) of GF which encodes the full mod-q cohomology ring H∗(GF, ℤ/q) and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when q = p is an odd prime, F(3) is the compositum of all Galois extensions E of F such that Gal(E/F) is isomorphic to {1}, ℤ/p or to the nonabelian group H_p³ of order p³ and exponent p.