Open Problems on Central Simple Algebras — Problem 9

Definitions and mathematical statement

Let $F$ be a field, and let $\operatorname{Br}(F)$ denote its Brauer group: the group of Brauer equivalence classes of finite-dimensional central simple $F$-algebras, with multiplication induced by tensor product. Two such algebras are Brauer equivalent if suitable matrix algebras over them are isomorphic as $F$-algebras.

A cyclic algebra of degree $4$ over $F$ is an algebra $(L/F,\sigma,a)$, where $L/F$ is a cyclic Galois extension of degree $4$, $\sigma$ generates $\operatorname{Gal}(L/F)$, and $a\in F^\times$. It has underlying vector space $\bigoplus_{j=0}^{3}Lu^j$, with multiplication specified by $u^4=a$ and $u\ell=\sigma(\ell)u$ for $\ell\in L$. Define \[ C_4(F)=\left\langle [(L/F,\sigma,a)] : L/F\text{ cyclic Galois of degree }4,\ \sigma\text{ a generator},\ a\in F^\times\right\rangle \subseteq\operatorname{Br}(F). \] Here generation means generation as a subgroup, so its elements are finite products of the indicated classes and their inverses.

The supplied Problem 9 asks whether, for every field $F$ satisfying \[ \operatorname{char}(F)\ne 2,\qquad -1\notin(F^\times)^2, \] one has \[ \operatorname{Br}_p[4](F)=C_4(F). \] The second hypothesis is the supplied condition $\sqrt{-1}\notin F$. The notation $\operatorname{Br}_p[4](F)$ is retained exactly from the record, but neither this notation nor the subscript $p$ is defined there. Consequently, the intended subgroup on the left remains unresolved, and the displayed question requires clarification before it is a fully specified mathematical statement. In particular, it has not been silently identified with the subgroup of classes annihilated by $4$.

The source introduces this as a question arising for composite $n$, with $n=4$ in the displayed instance. It also supplies the following related question, attributed to M. Mahdavi-Hezavehi: for every field $F$, does divisibility of $F^\times$ imply $\operatorname{Br}(F)=0$? Divisibility means that for every $a\in F^\times$ and every integer $m\geq1$, there exists $b\in F^\times$ with $b^m=a$. As background, the source states that this related question has an affirmative answer when $\operatorname{Br}(F)$ is generated by cyclic algebras, because these algebras split over such a field. It further states that the answer is affirmative in characteristic zero, since divisibility then supplies all roots of unity and the Merkurjev--Suslin theorem applies. These assertions are retained as supplied source qualifications, without independent verification.

Short English statement

For every field $F$ of characteristic different from $2$ that does not contain a square root of $-1$, the source asks whether $\operatorname{Br}_p[4](F)$ is generated by classes of cyclic algebras of degree $4$. The target notation is undefined in the supplied record, so its intended meaning requires review. The registry labels the problem open; that label is supplied curation, not an independently verified status assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800699$, exact source problem code catalogue entry, titled Open Problems on Central Simple Algebras --- Problem 9. The supplied record attributes the problem to Auel, Brussel, Garibaldi, and Vishne, with proposed year $2010$, and labels it open.
• Auel, Brussel, Garibaldi, and Vishne, Open Problems on Central Simple Algebras (2010), Problem 9. Supplied source URL: https://arxiv.org/abs/1006.3304. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not claims of access or verification for this draft.
• The related divisibility question is attributed in the supplied text to M. Mahdavi-Hezavehi. The same text invokes the Merkurjev--Suslin theorem for its characteristic-zero qualification, but supplies no separate bibliographic reference for either attribution.