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ON GENERALISED LEGENDRE MATRICES INVOLVING ROOTS OF UNITY OVER FINITE FIELDS
Published online by Cambridge University Press: 22 April 2024
Abstract
Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, Acta Arith. 115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting 
$a_1,\ldots ,a_{(q-1)/2}$ be all the nonzero squares in the finite field 
$\mathbb {F}_q$ containing q elements with 
$2\nmid q$, we give the explicit value of the determinant 
$D_{(q-1)/2}=\det [(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if 
$q=p$ is a prime greater than 
$3$, then 
$$ \begin{align*}\bigg(\frac{\det D_{(p-1)/2}}{p}\bigg)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4,\\ (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}\end{align*} $$
where 
$(\frac {\cdot }{p})$ is the Legendre symbol and 
$h(-p)$ is the class number of 
$\mathbb {Q}(\sqrt {-p})$.
- Type
- Research Article
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- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
This work was supported by the Natural Science Foundation of China (Grant No. 12101321).
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