Gauss Sum of Index 4: (1) Cyclic Case

Let p be a prime, m . 2, and (m, p(p . 1)) = 1. In this paper, we will calculate explicitly the Gauss sum G(.)=.x.F.q.(x).T(x)pin the case of [(./m.)* : .p.] = 4, and .1 . .p., where q = p f, f=.(m)4, . is a multiplicative character of F q with order m, and T is the trace map for F q /F p . Under the assumptions [(./m.)* : .p.] = 4 and .1 . .p., the decomposition field of p in the cyclotomic field .(. m ) is an imaginary quartic (abelian) field. And G (.) is an integer in K. We deal with the case where K is cyclic in this paper and leave the non.cyclic case to the next paper.