# Determination of quartic Gauss sums

Typically, the Gauss sum over $\mathbb{F}_p$ of order $k$ means the quantity $$\sum_{n=0}^{p-1} e^{2\pi i n^k/p}.$$ In the book Gauss and Jacobi Sums of Berndt, Evans and Williams, a more general sum is examined in the case $k=2$: namely for an arbitrary integer $m$, $$\sum_{n=0}^{p-1} e^{2\pi i m n^2/p},$$ where the previous sum is obtained by setting $m=1$. The two sums differ by a factor of a Legendre symbol. Specifically, we have $$\sum_{n=0}^{p-1} e^{2\pi i m n^2/p} = \left(\frac{m}{p}\right)\sum_{n=0}^{p-1} e^{2\pi i n^2/p}.$$ It seems that the authors consider no such generalization for any other values of $k$. In particular, I am interested in the case $k=4$. Is anything known about the quartic sums $$\sum_{n=0}^{p-1} e^{2\pi i m n^4/p},$$ for $m\not\equiv 1\pmod{p}$? A lovely determination is given when $m=1$, but I cannot find a source that has considered other cases. Any perspective on the matter is welcome.

• When there is an $a \in \mathbb{F}_p$ such that $a^4=m$, we would get $mn^4 = (an)^4$. Then the substitution $n \mapsto a^{-1}n$ would prove that the value of the sum is the same as when $m=1$. – Krijn Aug 8 '15 at 1:07