Quotients of absolute Galois groups which determine the entire Galois cohomology

Abstract

For prime power $q=p^d$ and a field $F$ containing a root of unity of order $q$ we show that the Galois cohomology ring $H^*(G_F,\dbZ/q)$ 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.

16 pages, Final version, To appear in Mathematische Annalen

References in corpus (1)

• An exact sequence for Milnor's K-theory with applications to quadratic forms