Galois groups and cohomological functors

Details Galois groups and cohomological functors Galois groups and cohomological functors

2011-03-08

arxiv.org/abs/1103.1508v1

Mathematics

Number Theory

AMS-LaTeX, 27 pages

Scientific paper

Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G=G_F$ its absolute Galois group. We find the minimal quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G$ which determines the full mod-$q$ cohomology ring $H^*(G,\mathbb{Z}/q)$. We prove that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E$ of $F$ such that $\mathrm{Gal}(E/F)$ is isomorphic to $\{1\}$, $\mathbb{Z}/p$ or to the nonabelian group $H_{p^3}$ of order $p^3$ and exponent $p$.

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