Some notes about power residues modulo prime - Diego Alejandro Mejía Guzmán; Yuki Kiriu (Revista Integración, 2022)
Let q be a prime. We classify the odd primes p ≠ q such that the equation x2 ≡ q (mod p) has a solution, concretely, we find a subgroup L4q of the multiplicative group U4q of integers relatively prime with 4q (modulo 4q) such that x2 ≡ q (mod p) has a solution iff p ≡ c (mod 4q) for some c ∈ L4q. Moreover, L4q is the only subgroup of U4q of half order containing −1.