Quadratic residue pattern and the Galois group of $\mathbb {Q}(\sqrt {a_{1}}, \sqrt {a_{2}}, \dots , \sqrt {a_{n}})$

Abstract:

Let $S= \{ a_{1}, a_{2}, \dots , a_{n} \}$ be a finite set of non-zero integers. R. Balasubramanian, F. Luca and R. Thangadurai [Proc. Amer. Math. Soc. 138 (2010), pp. 2283–2288] gave an exact formula for the degree of the multi-quadratic field $\mathbb {K}= \mathbb {Q}(\sqrt {a_{1}}, \sqrt {a_{2}}, \dots , \sqrt {a_{n}})$ over $\mathbb {Q}$. In this paper, we calculate the explicit structure of the Galois group $\operatorname {Gal}(\mathbb {K}/\mathbb {Q})$ in terms of its action on $\sqrt {a_{i}}$ for $1 \leq i \leq n$.