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Let $S= \{ a_{1}, a_{2}, \dots, a_{n} \}$ be a finite set of non-zerointegers. In \cite{KBAM21}, Karthick Babu and Anirban Mukhopadhyay calculatedthe explicit structure of the Galois group of multi-quadratic field$\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}})$ over$\mathbb{Q}$. For a positive integer $d \geq 3$, $\zeta_{d}$ denotes theprimitive $d$-th root of unity. In this paper, we calculate the explicitstructure of the Galois group of $\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots,\sqrt{a_{n}}, \zeta_{d})$ over $\mathbb{Q}$ in terms of its action on$\zeta_{d}$ and $\sqrt{a_{i}}$ for $1 \leq i \leq n$.

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