Open Problems on Central Simple Algebras — Problem 10

Definitions and mathematical statement

Let $F$ be a field. A central simple $F$-algebra is a finite-dimensional associative unital $F$-algebra that is simple and has center $F$. Its degree is $\deg(A)=\sqrt{\dim_F A}$. The Brauer group $\operatorname{Br}(F)$ consists of equivalence classes of central simple $F$-algebras, where $A$ and $B$ are equivalent if $M_r(A)\cong M_s(B)$ as $F$-algebras for some positive integers $r,s$. The group operation is induced by tensor product over $F$.

For a positive integer $n$, define the subgroup \[ H_n(F)=\langle [A]\in\operatorname{Br}(F): A\text{ is a central simple }F\text{-algebra with }\deg(A)=n\rangle. \] The supplied source asks, as a weaker question, \[ \operatorname{Br}_p[n](F)\stackrel{?}{=}H_n(F). \] This equality transcribes the question under the usual interpretation of generation inside the Brauer group. The supplied record does not define $\operatorname{Br}_p[n](F)$, specify the meaning or range of $p$, give the permitted values of $n$, or state any restrictions on $F$. The field and positive-integer conventions above define the right-hand side; they do not resolve the missing source hypotheses. Thus the display is an unresolved formulation template, not a fully quantified conjecture. The stronger question to which the source compares it is also absent.

Short English statement

The source asks whether the group denoted by $\operatorname{Br}_p[n](F)$ is generated by Brauer classes of central simple $F$-algebras of degree $n$. Its notation and parameter restrictions are missing from the supplied excerpt, so a complete quantified statement requires further source context.

Sources

• Local UnsolvedMath registry: supplied complete source record with permanent ID $800700$, exact source problem code catalogue entry, titled Open Problems on Central Simple Algebras --- Problem 10. The registry attributes the problem to Auel, Brussel, Garibaldi, and Vishne, with proposed year $2010$. Its open status is supplied registry metadata, not an independently verified mathematical or literature assessment.
• Auel, Brussel, Garibaldi, and Vishne, Open Problems on Central Simple Algebras (2010), Problem 10. Supplied source URL: https://arxiv.org/abs/1006.3304. The background records access on $2026\text{-}10\text{-}16$ and extraction method arxiv_latex; these are supplied provenance details, not a claim of independent source consultation.