Some notes about power residues modulo prime

Abstract

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 L_4q of the multiplicative group U_4q 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 ∈ L_4q. Moreover, L_4q is the only subgroup of U_4q of half order containing −1.

Considering the ring Z[√2], for any odd prime p it is known that the equation x² ≡ 2 (mod p) has a solution iff the equation x² −2y² = p has a solution in the integers. We ask whether this can be extended in the context of Z[ⁿ√2] with n ≥2, namely: for any prime p ≡ 1 (mod n), is it true that xⁿ ≡ 2 (mod p) has a solution iff the equation D²_n(x₀, . . . , x_n−1) = p has a solution in the integers? Here D²_n(x̄) represents the norm of the field extension Q(ⁿ√2) of Q. We solve some weak versions of this problem, where equality with p is replaced by 0 (mod p) (divisible by p), and the “norm" D^r_n(x̄) is considered for any r ∈ Z in the place of 2.