On the explicit Galois group of \(\mathbb {Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots , \sqrt{a_{n}}, \zeta _{d})\) over \(\mathbb {Q}\)

Let \(S= \{ a_{1}, a_{2}, \dots , a_{n} \}\) $S=\{a_1,a_2,\cdots ,a_n\}$ be a finite set of non-zero integers. In [5], Karthick Babu and Anirban Mukhopadhyay calculated the explicit structure of the Galois group of multi-quadratic field \(\mathbb {Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots , \sqrt{a_{n}})\) $\mathbb{Q}(\sqrt{a_1},\sqrt{a_2},\cdots ,\sqrt{a_n})$ over \(\mathbb {Q}\) $\mathbb{Q}$ . For a positive integer \(d \geqslant 3\) $d⩾3$ , \(\zeta _{d}\) ${\zeta }_d$ denotes the primitive d-th root of unity. In this paper, we calculate the explicit structure of the Galois group of \(\mathbb {Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots , \sqrt{a_{n}}, \zeta _{d})\) $\mathbb{Q}(\sqrt{a_1},\sqrt{a_2},\cdots ,\sqrt{a_n},{\zeta }_d)$ over \(\mathbb {Q}\) $\mathbb{Q}$ in terms of its action on \(\zeta _{d}\) ${\zeta }_d$ and \(\sqrt{a_{i}}\) $\sqrt{a_i}$ for \(1 \leqslant i \leqslant n\) $1⩽i⩽n$ .