Let
S
=
{
a
1
,
a
2
,
…
,
a
ℓ
}
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
Q
(
a
1
,
a
2
,
…
,
a
ℓ
)
\mathbb {Q}(\sqrt {a_1}, \sqrt {a_2},\ldots , \sqrt {a_\ell })
over
Q
\mathbb {Q}
. To do this, we compute the relative density of the set of prime numbers
p
p
for which all the
a
i
a_i
’s are simultaneously quadratic residues modulo
p
p
in two ways.