Abstract

KIRIU, YUKI  and  MEJIA, DIEGO A.. Some notes about power residues modulo prime. Integración - UIS [online]. 2022, vol.40, n.1, pp.1-23.  Epub Aug 26, 2022. ISSN 0120-419X.  https://doi.org/10.18273/revint.v40n1-2022001.

Let q be a prime. We classify the odd primes p ≠ q such that the equation x2 ≡ 2 (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.

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 x n ≡ 2 (mod p) has a solution iff the equation D 2 n (x 0 ,..., x n-1 ) = p has a solution in the integers? Here D 2 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 2 n (x̄) is considered for any r ∈ ℤ in the place of 2.

Keywords : Power residues modulo prime; quadratic residues; Legendre symbol; norms of field extensions; irreducible polynomials.

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