Quotients of absolute Galois groups which determine the entire Galois cohomology

Details Quotients of absolute Galois groups which determine the entire Galois cohomology Quotients of absolute Galois groups which determine the entire Galois cohomology

2009-05-09

arxiv.org/abs/0905.1364v6

Mathematics

Group Theory

16 pages, Final version, To appear in Mathematische Annalen

Scientific paper

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.

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