Galois groups and cohomological functors
Abstract
Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod$q$ cohomology ring $H^*(G_F,\mathbb{Z}/q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown 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 $p$.
 Publication:

arXiv eprints
 Pub Date:
 March 2011
 arXiv:
 arXiv:1103.1508
 Bibcode:
 2011arXiv1103.1508E
 Keywords:

 Mathematics  Number Theory;
 Mathematics  KTheory and Homology;
 12G05 (Primary) 12E30 (Secondary)
 EPrint:
 AMSLaTeX, 29 pages. To appear in the Transactions of the American Mathematical Society