Some notes about power residues modulo prime

Abstract

Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively prime with $4q$ (modulo $4q$) such that $x^2\equiv q\pmod{p}$ has a solution iff $p\equiv c\pmod{4q}$ for some $c\in\mathbb{L}_{4q}$. Moreover, $\mathbb{L}_{4q}$ is the only subgroup of $\mathbb{U}_{4q}$ of half order containing $-1$. Considering the ring $\mathbb{Z}[\sqrt{2}]$, for any odd prime $p$ it is known that the equation $x^2\equiv 2\pmod{p}$ has a solution iff the equation $x^2-2y^2=p$ has a solution in the integers. We ask whether this can be extended in the context of $\mathbb{Z}[\sqrt[n]{2}]$ with $n\geq 2$, namely: for any prime $p\equiv 1\pmod{n}$, is it true that $x^n\equiv 2\pmod{p}$ has a solution iff the equation $D^2_n(x_0,\ldots,x_{n-1})=p$ has a solution in the integers? Here $D^2_n(\bar{x})$ represents the norm of the field extension $\mathbb{Q}(\sqrt[n]{2})$ of $\mathbb{Q}$. We solve some weak versions of this problem, where equality with $p$ is replaced by $0\pmod{p}$ (divisible by $p$), and the "norm" $D^r_n(\bar{x})$ is considered for any $r\in\mathbb{Z}$ in the place of $2$.