On the structure of the Brauer group of fields

This paper is devoted to the study of the structure of the Brauer group of an arbitrary field. It is proved that, for any odd prime $p$ different from the characteristic of the field $F$, the subgroup $_q\mathrm{Br}(F)$ of elements of order $q=p^n$ in the Brauer group of $F$ is generated by the images of the cyclic algebras $A_\xi(x,y)$ under the corestriction map $_q\mathrm{Br}(F(\xi_q))\to{_q\mathrm{Br}}(F)$. As a corollary it is shown that $_q\mathrm{Br}(F)$ is generated by elements whose index is bounded by $q^{q/p}$.
A representation of the $p$-component $\mathrm{Br}(F)\{p\}$ of the Brauer group by means of generators and relations is obtained, and the specialization homomorphism $\mathrm{Br}(T)\{p\}\to\mathrm{Br}(K)\{p\}$, where $T$ is a local algebra and $K$ is the residue field, is shown to be surjective. Similar results are obtained in the case $p=2$.
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