Open Problems on Central Simple Algebras — Problem 14

Definitions and mathematical statement

For a finitely generated field extension $K/F$, call $K/F$ rational if $K\cong_F F(t_1,\ldots,t_r)$ for some integer $r\geq 0$, where the $t_i$ are algebraically independent over $F$. Call it stably rational if there exist integers $r,s\geq 0$ and algebraically independent indeterminates such that \[ K(u_1,\ldots,u_r)\cong_F F(v_1,\ldots,v_s). \] Call it retract rational if there exist a finitely generated integral $F$-subalgebra $R\subseteq K$ with $\operatorname{Frac}(R)=K$, an integer $s\geq 0$, a nonzero polynomial $f\in F[t_1,\ldots,t_s]$, and unital $F$-algebra homomorphisms \[ R\xrightarrow{\alpha}F[t_1,\ldots,t_s][1/f]\xrightarrow{\beta}R, \qquad \beta\circ\alpha=\operatorname{id}_R. \] Two extensions $K/F$ and $L/F$ are stably $F$-isomorphic if $K(u_1,\ldots,u_r)\cong_F L(v_1,\ldots,v_s)$ for some finite sets of algebraically independent indeterminates.

The supplied source asks whether each of the extensions \[ Z(F,n)/F,\qquad Z_m(F,n)/F,\qquad Z_H(F,n)/F \] is rational, stably rational, or retract rational, with the three properties considered separately. However, the supplied record does not define these fields, specify the hypotheses on $F$, or give the admissible values and meaning of the parameters $n,m,H$. Thus this is a conditional statement of the intended rationality questions: the target extensions and their parameter ranges must be recovered from the cited source before a fully specified quantified problem can be stated. The definitions above do not supply those missing constructions or assert that the displayed extensions satisfy their prerequisites.

As background, the source reports a result attributed to Katsylo and Schofield: for admissible indices satisfying $n=ab$ and $\gcd(a,b)=1$, the field $Z(F,n)$ is stably isomorphic to the compositum $Z(F,a)Z(F,b)$. The supplied record does not specify the embeddings or ambient field used for this compositum, nor the full hypotheses of that result. It also reports that similar statements are possible for $Z_m(F,n)$ and $Z_H(F,n)$, without stating them precisely. The source qualifies the resulting reduction to prime-power indices as applying often, but not always.

Short English statement

The problem asks whether the source's fields $Z(F,n)$, $Z_m(F,n)$, and $Z_H(F,n)$ are rational, stably rational, or retract rational over $F$. Their constructions and parameter restrictions are absent from the supplied record, so the intended scope remains unresolved. The registry labels the problem open; that label is supplied metadata, not an independently verified status assessment.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800704$, exact source problem code catalogue entry, title Open Problems on Central Simple Algebras --- Problem 14. The record attributes the entry 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 14. Supplied source URL: https://arxiv.org/abs/1006.3304. The provenance records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance claims, not independently verified access or source checks.
• The question carries the supplied reference label procesi:67, p. 240. No full bibliographic entry for this label is supplied.
• The background attributes the relatively prime factorization result to Katsylo, reference label Katsylo, and Schofield, reference label Schofield. Full bibliographic entries and precise theorem hypotheses are not supplied.