On the structure of the Brauer group of fields

A. S. Merkur'ev

Izvestiya. Mathematics , Tome 27 (1986) no. 1, pp. 141-157

Résumé

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 $p=2$. Bibliography: 20 titles.