On the explicit Galois group of $\mathbb{Q}\Big(a_{1}^{\frac{1}{m}}, a_{2}^{\frac{1}{m}}, \dots, a_{n}^{\frac{1}{m}}, \zeta_{m}\Big)$ over $\mathbb{Q}$

Let $m>1$ be an integer, and $\zeta_m$ denote a primitive $m$-th root of unity. For any finite set $S=\{a_{1},a_{2},\dots,a_{n}\}$ and arbitrary choice of $m$-th roots of unity $\zeta_m^{r_i}$, $0\leq r_i < m$ for $i=1,\dots,n$, we study the density of primes $\mathbf{p}$ of $\mathbb{Q}(\zeta_m)$ such that the $m$-th power residue symbol $\left(\frac{a_i}{\mathbf{p}}\right)_m=\zeta_m^{r_i}$. We calculate the explicit structure of the Galois group $\operatorname{Gal}{\mathbb{Q}(a_1^{\frac{1}{m}},\ldots,a_n^{\frac{1}{m}},\zeta_m)/\mathbb{Q}(\zeta_{m})}$ in terms of its action on $a_{i}^{\frac{1}{m}}$ for $1\leq i\leq n$.