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On the descending central sequence of absolute Galois groups
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 6, December 2011
- pp. 1503-1532
- 10.1353/ajm.2011.0041
- Article
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Let $p$ be an odd prime number and $F$ a field containing a primitive
$p$th root of unity. We prove a new restriction on the group-theoretic
structure of the absolute Galois group $G_F$ of $F$. Namely, the third
subgroup $G_F^{(3)}$ in the descending $p$-central sequence of $G_F$ is
the intersection of all open normal subgroups $N$ such that $G_F/N$ is
$1$, ${\Bbb Z}/p^2$, or the extra-special group $M_{p^3}$ of order $p^3$
and exponent $p^2$.