On Sign Ambiguities of Multiplicative Relations Between Gauss Sums

Let m> 1 be an integer and put ζm = e2πi/m. We denote by Km = Q(ζm) the m-th cyclotomic field, whose integer ring will be denoted by OKm. Let p be a prime number such that p ≡ 1 (mod m). Throughout the present paper we fix a prime ideal P of OKm dividing p. Since p splits completely in Km, the residue fieldOKm/P can be identified wit