Open Problems on Central Simple Algebras — Problem 21

Definitions and mathematical statement

Let $F$ be a field and let $G$ be an absolutely almost simple linear algebraic group over $F$: thus $G$ is connected and semisimple, and over an algebraic closure of $F$ it has no nontrivial proper connected normal algebraic subgroup. For an extension $L/F$, write $G_L=G\times_F L$. The group $G_L$ is anisotropic over $L$ if it contains no nontrivial $L$-split torus, that is, no $L$-subgroup isomorphic to $(\mathbb G_{m,L})^r$ for any integer $r\geq 1$.

The supplied statement uses $S(G)$ as a set of primes but does not define that set. Retaining this unresolved source notation, the question is: whenever $G$ is anisotropic over $F$, is it true that, for every finite field extension $K/F$, \[ \left(\forall p\in S(G),\ p\nmid [K:F]\right) \quad\Longrightarrow\quad G_K\text{ is anisotropic over }K? \] Here $[K:F]=\dim_F K$. The source imposes no restriction on the characteristic of $F$ and does not require $K/F$ to be separable. A complete formulation requires the intended definition of $S(G)$; it is not being quantified over arbitrary sets of primes.

Short English statement

Does every anisotropic absolutely almost simple group remain anisotropic after every finite field extension whose degree is divisible by none of the primes in the source's set $S(G)$? The supplied record does not define $S(G)$, so this draft retains an unresolved parameter.

Sources

• Local UnsolvedMath registry, permanent ID $800711$, exact source problem code catalogue entry, titled Open Problems on Central Simple Algebras --- Problem 21.
• Auel, Brussel, Garibaldi, and Vishne, Open Problems on Central Simple Algebras (2010), Problem 21. Supplied source URL: https://arxiv.org/abs/1006.3304. The registry attributes the problem to these authors and records proposed year 2010.
• The supplied provenance records access on 2026-10-16 and extraction from arXiv LaTeX. These are supplied provenance claims, not a claim of independent source consultation. The registry labels the problem open; that label is not an independently verified assessment of its mathematical status.