Quotients of absolute Galois groups which determine the entire Galois cohomology

Abstract

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.