A note on some diagonal cubic equations over finite fields

Abstract

<p>Let a prime $ p\equiv 1(\text{mod}3) $ and $ z $ be non-cubic in $ \mathbb{F}_p $. Gauss proved that the number of solutions of equation</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x_1^3+x_2^3+zx_3^3 = 0 $\end{document} </tex-math></disp-formula></p><p>in $ \mathbb{F}_p $ was $ p^2+\frac{1}{2}(p-1)(9d-c) $, where $ c $ was uniquely determined and $ d $, except for the sign, was defined by</p><p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ 4p = c^2+27d^2,\ \ c\equiv 1(\text{mod}3). $\end{document} </tex-math></disp-formula></p><p>In 1978, Chowla, Cowles, and Cowles determined the sign of $ d $ for the case of 2 being non-cubic in $ \mathbb{F}_p $. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field $ \mathbb{F}_q $ with $ q = p^k $, $ p\equiv 1(\text{mod}3) $, and determined the sign of $ d $ for the case of 3 being non-cubic.</p>