 The cubic exponential sum "K"("n","p") defined by

is easily seen to be a linear combination of the Kummer sums.In fact it is 3"P" where "P" is one of the Gaussian periods for the subgroup of index 3 in the residues mod "p", under multiplication, while the Gauss sums are linear combinations of the "P" with cube roots of unity as coefficients.However it is the Gauss sum for which the algebraic properties hold.