The Ramanujan Journal ( IF 0.7 ) Pub Date : 2020-07-23 , DOI: 10.1007/s11139-020-00277-8 Li-Yuan Wang , Hao Pan
Let p be an odd prime and let \({\mathbb {F}}_p\) denote the finite field with p elements. Suppose that g is a primitive root of \({\mathbb {F}}_p\). Define the permutation \(\tau _g:\,{\mathcal {H}}_p\rightarrow {\mathcal {H}}_p\) by
$$\begin{aligned} \tau _g(b):={\left\{ \begin{array}{ll} g^b&{}\text {if }g^b\in {\mathcal {H}}_p,\\ -g^b&{}\text {if }g^b\not \in {\mathcal {H}}_p,\\ \end{array}\right. } \end{aligned}$$for each \(b\in {\mathcal {H}}_p\), where \({\mathcal {H}}_p=\{1,2,\ldots ,(p-1)/2\}\) is viewed as a subset of \({\mathbb {F}}_p\). In this paper, we investigate the sign of \(\tau _g\). For example, if \(p\equiv 5\ (\mathrm{{mod}}\ 8)\), then
$$\begin{aligned} (-1)^{|\tau _g|}=(-1)^{\frac{1}{4}(h(-4p)+2)} \end{aligned}$$for every primitive root g, where \(h(-4p)\) is the class number of the ring of integers of the imaginary quadratic field \({\mathbb {Q}}(\sqrt{-4p})\).
中文翻译：
关于原始根的$$ \ pmb {\ mathbb {F}} _ p $$ F p的一些置换
令p为奇数质数，令\（{\ mathbb {F}} _ p \）表示具有p个元素的有限域。假设g是\（{\ mathbb {F}} _ p \）的原始根。限定置换\（\ tau蛋白_g：\ {\ mathcal {H}} _ p \ RIGHTARROW {\ mathcal {H}} _ p \）由
$$ \ begin {aligned} \ tau _g（b）：= {\ left \ {\ begin {array} {ll} g ^ b＆{} \ text {if} g ^ b \在{\ mathcal {H}}中_p，\\ -g ^ b＆{} \ text {if} g ^ b \ not \ in {\ mathcal {H}} _ p，\\ \ end {array} \ right中。} \ end {aligned} $$对于每个\（b \ in {\ mathcal {H}} _ p \），其中\（{\ mathcal {H}} _ p = \ {1,2，\ ldots，（p-1）/ 2 \} \）被视为\（{\ mathbb {F}} _ p \）的子集。在本文中，我们研究\（\ tau _g \）的符号。例如，如果\（p \ equiv 5 \（\ mathrm {{mod}} \ 8）\），则
$$ \ begin {aligned}（-1）^ {| \ tau _g |} =（-1）^ {\ frac {1} {4}（h（-4p）+2）} \ end {aligned} $ $对于每个原始根g，其中\（h（-4p）\）是虚二次域\（{\ mathbb {Q}}（\ sqrt {-4p}）\）的整数环的类号。
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