An application of Galois theory to the optimal control

Define the Zelikin-Lokutsievskiy polynom $f_n(x)$ with integer coefficients of degree $n-1$ as follows
$$ xf_n(x^2)=\mathrm{Im}\,(ix+1)\ldots(ix+2n). $$

We show the irreducibility of $f_{(q-1)/2}(x)$ over $\mathbb Q$ for any prime $q>3$. We calculate the Galois group of the polynom $f_n(x)$, when the numbers $p=n-1$, $q=2n+1$, $r=2n+7$ are prime and $889$ is not a square modulo $r$. We also show under irreducibility hypothesis of the polynom $f_{p+1}(x)$ over $\mathbb Q$ for almost all primes $p$ that there exists an infinite sequence of natural $n$, for which $A_{n-1}$ is embeddable into $\mathrm{Gal}_{\mathbb Q}(f_n(x))$.
An example: for any natural $k<808$ there exists an optimal control problem, the optimal control of which throws a dense winding of the $k$-dimensional torus in finite time.