Squares in $\mathbb{F}_{p^2}$ and permutations involving primitive roots

Abstract

Let $p=2n+1$ be an odd prime, and let $\zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. We let $g\in\mathbb{Z}_p[\zeta_{p^2-1}]$ be a primitive root modulo $p\mathbb{Z}_p[\zeta_{p^2-1}]$ with $g\equiv \zeta_{p^2-1}\pmod {p\mathbb{Z}_p[\zeta_{p^2-1}]}$. Let $\Delta\equiv3\pmod4$ be an arbitrary quadratic non-residue modulo $p$ in $\mathbb{Z}$. By the Local Existence Theorem we know that $\mathbb{Q}_p(\sqrt{\Delta})=\mathbb{Q}_p(\zeta_{p^2-1})$. For all $x\in\mathbb{Z}[\sqrt{\Delta}]$ and $y\in\mathbb{Z}_p[\zeta_{p^2-1}]$ we use $\bar{x}$ and $\bar{y}$ to denote the elements $x\mod p\mathbb{Z}[\sqrt{\Delta}]$ and $y\mod p\mathbb{Z}_p[\zeta_{p^2-1}]$ respectively. If we set $a_k=k+\sqrt{\Delta}$ for $0\le k\le p-1$, then we can view the sequence $$S := \overline{a_0^2}, \cdots, \overline{a_0^2n^2}, \cdots,\overline{a_{p-1}^2}, \cdots, \overline{a_{p-1}^2n^2}\cdots, \overline{1^2}, \cdots,\overline{n^2}$$ as a permutation $\sigma$ of the sequence $$S^* := \overline{g^2}, \overline{g^4}, \cdots,\overline{g^{p^2-1}}.$$ We determine the sign of $\sigma$ completely in this paper.