21.77 (2026)

Let $d$ be an integer that is not divisible by $n$-th powers of primes, let $x^n - d$ be an irreducible polynomial over $\mathbb{Q}$, let $\theta = \sqrt[n]{d}$, and let $K = \mathbb{Q}(\theta)$ be the radical extension of degree $n$ of the field $\mathbb{Q}$. The multiplicative group $K^*$ of the field $K$ is canonically embedded into the group $\text{Aut}_{\mathbb{Q}}(K)$ of all invertible $\mathbb{Q}$-linear mappings of the $\mathbb{Q}-space$ $K$; let $T$ be the image of $K^*$ under this embedding. In the natural basis $1, \theta, \theta^2, \dots, \theta^{n-1}$ of the $\mathbb{Q}-space$ $K$ the group $\text{Aut}_{\mathbb{Q}}(K)$ corresponds to $G = \text{GL}(n, \mathbb{Q})$, and the subgroup $T$ to a subgroup $T(d)$ (unsplit maximal torus). Every subgroup $H$ of $G$ containing $T(d)$ and some one-dimensional transformation is rich in elementary transvections (V. A. Koibaev, St. Petersbg. Math. J., 21, no. 5 (2010), 731–742) and thus defines a net $\sigma = \sigma(H)$ (V. A. Koibaev, A. V. Shilov, J. Math. Sci. New York, 171, no. 3 (2010), 380–385). Let $E(s)$ denote the subgroup generated by all transvections in the net group $G(\sigma)$. Is it true that $H \leqslant N_G(E(\sigma))$?