Open Problems on Central Simple Algebras — Problem 7

Definitions and mathematical statement

Let $F$ be a field. Its Brauer group $\operatorname{Br}(F)$ consists of Brauer equivalence classes of finite-dimensional central simple $F$-algebras, with addition induced by tensor product. For a positive integer $n$, write \[ \operatorname{Br}(F)[n]=\{\alpha\in\operatorname{Br}(F):n\alpha=0\}. \] A cyclic algebra of degree $d$ over $F$ is an algebra $(L/F,\sigma,a)$, where $L/F$ is a cyclic Galois extension of degree $d$, $\sigma$ generates its Galois group, and $a\in F^\times$. It is generated by $L$ and an element $u$, subject to $u^d=a$ and $u\ell=\sigma(\ell)u$ for all $\ell\in L$.

The source asks whether its subgroup $\operatorname{Br}_p[n](F)$ is generated by classes of cyclic algebras whose degrees divide $n$. With the source's notation retained, the requested assertion is \[ \operatorname{Br}_p[n](F) \stackrel{?}{=} \left\langle [A]\in\operatorname{Br}(F): A\text{ is a cyclic }F\text{-algebra and }\deg(A)\mid n \right\rangle. \] Here generation means taking finite sums and additive inverses in the Brauer group. The supplied record does not define the subscript $p$, specify its relation to $n$ or $\operatorname{char}(F)$, or state the intended restrictions on $F$ and $n$. Consequently, the displayed equality records the question but does not supply a fully specified universally quantified conjecture. In particular, identifying $\operatorname{Br}_p[n](F)$ with $\operatorname{Br}(F)[n]$ would require information absent from the record.

Short English statement

Can every class in the source's $n$-torsion subgroup $\operatorname{Br}_p[n](F)$ be generated by classes of cyclic algebras of degree dividing $n$? The meaning of $p$ and the intended field and integer hypotheses remain unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $800697$, source problem code catalogue entry, titled "Open Problems on Central Simple Algebras --- Problem 7."
• Auel, Brussel, Garibaldi, and Vishne, Open Problems on Central Simple Algebras (2010), Problem 7. Supplied source URL: https://arxiv.org/abs/1006.3304. The record reports access on 2026-10-16 and extraction from arXiv LaTeX; these are supplied provenance details, not independently verified here.
• The statement includes the attribution [Albert:cyclics, p. 126]. The supplied record does not resolve this bibliographic key to a complete reference.