Quotients of absolute Galois groups which determine the entire Galois cohomology

摘要

For a prime power q = p d and a field F containing a root of unity of order q we show that the Galois cohomology ring ${H^*(G_F,mathbb{Z}/q)}$ is determined by a quotient ${G_F^{[3]}}$ of the absolute Galois group G F related to its descending q-central sequence. Conversely, we show that ${G_F^{[3]}}$ is determined by the lower cohomology of G F . This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.