On the exact degree of ℚ(√𝕒₁,√𝕒₂,…,√𝕒_{ℓ}) over ℚ

Abstract

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.