Gauss sums of orders six and twelve (Q2739248)

For an integer \(k>1\) and a prime \(p \equiv 1 \bmod k\), define the Gauss sum \(G(k)\) by NEWLINE\[NEWLINE G(k)= \sum_{n=0}^{p-1} ~\exp ( 2 \pi i n^k /p). NEWLINE\]NEWLINE The author presents a complete evaluation of \(G(12)\), in terms of the known quantities \(G(3)\) and \(G(4)\). Moreover, he gives a new formulation of \(G(6)\) in terms of \(G(3)\) which is more elegant than that obtained previously. The problem of determining \(G(8)\) in terms of \(G(4)\) is still open and appears to be difficult.