On the explicit Galois group of $\mathbb{Q}(\sqrt{a_1},\sqrt{a_2},\dots ,\sqrt{a_n},{\zeta }_d)$ over $\mathbb{Q}$

Abstract

Let $S=\{a_1,a_2,\dots ,a_n\}$ be a finite set of non-zero integers. In \cite{KBAM21}, Karthick Babu and Anirban Mukhopadhyay calculated the explicit structure of the Galois group of multi-quadratic field $\mathbb{Q}(\sqrt{a_1},\sqrt{a_2},\dots ,\sqrt{a_n})$ over $\mathbb{Q}$. For a positive integer $d\ge 3$, ${\zeta }_d$ denotes the primitive $d$-th root of unity. In this paper, we calculate the explicit structure of the Galois group of $\mathbb{Q}(\sqrt{a_1},\sqrt{a_2},\dots ,\sqrt{a_n},{\zeta }_d)$ over $\mathbb{Q}$ in terms of its action on ${\zeta }_d$ and $\sqrt{a_i}$ for $1\le i\le n$.