Open Problems on Central Simple Algebras — Problem 15

Definitions and mathematical statement

Let $S_9$ be the symmetric group on nine letters, and let $H\subset S_9$ be a transitive subgroup isomorphic to $C_3\times C_3$, where $C_3$ denotes the cyclic group of order three. Transitivity means that for any two letters there is an element of $H$ carrying the first to the second.

Write $K=Z_H(F,9)$ for the field named in the source. The supplied record does not define the construction $Z_H(F,n)$, nor does it specify the admissible base fields $F$. Consequently, this notation cannot be replaced here by a fully specified field construction, and the intended quantification over $F$ remains unresolved.

For an extension $K/F$, say that $K$ is rational over $F$ if $K\cong F(t_1,\ldots,t_r)$ as an $F$-field for some integer $r\geq 0$, with the $t_i$ algebraically independent over $F$. Say that $K$ is stably rational over $F$ if, for some integer $s\geq 0$ and indeterminates $u_1,\ldots,u_s$ algebraically independent over $K$, the field $K(u_1,\ldots,u_s)$ is rational over $F$. Say that $K$ is retract rational over $F$ if there exist a finitely generated $F$-subalgebra $A\subset K$ with $\operatorname{Frac}(A)=K$, an integer $r\geq 0$, a nonzero polynomial $f\in F[t_1,\ldots,t_r]$, and $F$-algebra homomorphisms \[ A\xrightarrow{\iota}F[t_1,\ldots,t_r][1/f]\xrightarrow{\rho}A, \qquad \rho\circ\iota=\operatorname{id}_A. \]

Problem (conditional on the missing source definitions and hypotheses). Determine whether $Z_H(F,9)$, for the transitive subgroup $H\cong C_3\times C_3\subset S_9$, is rational over $F$, stably rational over $F$, or retract rational over $F$. Determine these properties separately; the source does not present them as mutually exclusive alternatives. A complete formulation requires the definition of $Z_H(F,n)$ and the intended hypotheses on $F$.

Supplied context. The record reports that $Z_2(F,4)$ is stably rational by Saltman, that $Z_2(F,8)$ is retract rational by Saltman--Tignol, and that $Z_2(F,8)$ is stably rational by Beneish. The notation $Z_m(F,n)$ and the hypotheses for these reports are not supplied. It also states that $Z_H(F,n)$ is rational when $H\subset S_n$ is cyclic and transitive and $F$ contains a primitive $n$-th root of unity. It mentions further results for general $F$, for $n$ not of the form $8m$, and concerning retract rationality, without giving their precise statements. These are supplied contextual claims, not independently verified results. The root-of-unity hypothesis in the cyclic example does not specify the intended hypotheses for the noncyclic group in this problem.

Short English statement

For the transitive permutation group $C_3\times C_3\subset S_9$, determine the rationality, stable rationality, and retract rationality over $F$ of the source's field $Z_H(F,9)$. The supplied record labels the problem open, but omits the field construction and the permitted base fields, so this statement remains conditional and requires mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800705$, exact source problem code catalogue entry, title Open Problems on Central Simple Algebras --- Problem 15. The registry attributes the item to Auel, Brussel, Garibaldi, and Vishne, with proposed year 2010, and supplies the status label "open." This label is supplied curation, not an independently verified current assessment.
• Auel, Brussel, Garibaldi, and Vishne, Open Problems on Central Simple Algebras (2010), Problem 15. Supplied source URL: https://arxiv.org/abs/1006.3304. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are provenance supplied with the record, not a claim of independent source access.
• Contextual references retained as supplied: Saltman [Salt:symp], for stable rationality of $Z_2(F,4)$; Saltman--Tignol [SaltTig], for retract rationality of $Z_2(F,8)$; Beneish [Beneish:28], for stable rationality of $Z_2(F,8)$. Complete bibliographic entries and their hypotheses are absent from the supplied record.