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 $g\equiv {\zeta }_{p^2-1}(\mathrm{mod}p{\mathbb{Z}}_p[{\zeta }_{p^2-1}])$. Let $\Delta \equiv 3(\mathrm{mod}4)$ 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 $x\mathrm{mod}p\mathbb{Z}[\sqrt{\Delta }]$ and $y\mathrm{mod}p{\mathbb{Z}}_p[{\zeta }_{p^2-1}]$ respectively. If we set $a_k=k+\sqrt{\Delta }$ for $0\le k\le p-1$, then we can view the sequence $$S:=\overline{a_0^2},\cdots ,\overline{a_0^2{n}^2},\cdots ,\overline{a_{p-1}^2},\cdots ,\overline{a_{p-1}^2{n}^2}\cdots ,\overline{1^2},\cdots ,\overline{n^2}$$ as a permutation $\sigma$ of the sequence $$S^{\ast }:=\overline{g^2},\overline{g^4},\cdots ,\overline{g^{p^2-1}}.$$ We determine the sign of $\sigma$ completely in this paper.

References in corpus (1)

• Applications of Lerch's theorem and permutations concerning quadratic residues