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

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. [ABSTRACT FROM AUTHOR]