Galois theory, the classification of finite simple groups and a dense winding of a torus

The Galois group of the Zelikin-Lokutsievskii polynomial is studied. It is established that, in the generalized Fuller problem, for any positive integer $k\leqslant 249\,994\,914$ there is an optimal control going along a dense winding of the $k$-dimensional torus in finite time. In the generalized Fuller problem, under the assumption that the Zelikin-Lokutsievskiy polynomials are irreducible over the field of rational numbers for almost all prime powers it is shown that there is an optimal control passing along a dense winding of the torus of any preassigned dimension in finite time. Many examples are considered.
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