On the exact degree of $\mathbb{Q}(\sqrt{a_1}, \sqrt{a_2},\ldots, \sqrt{a_\ell})$ over $\mathbb{Q}$

Let $S =\{a_1, a_2, \ldots, a_\ell\}$ be a finite set of non-zero integers. In this paper, we give an exact formula for the degree of the multi-quadratic field $\mathbb{Q}(\sqrt{a_1}, \sqrt{a_2},\ldots, \sqrt{a_\ell})$ over $\mathbb{Q}$. To do this, we compute the relative density of the set of prime numbers $p$ for which all the $a_i$'s are simultaneously quadratic residues modulo $p$ in two ways.