Infinite class of new sign ambiguities

AbstractIn 1934, two kinds of multiplicative relations, the norm and the Davenport–Hasse relations, between Gauss sums, were known. In 1964, H. Hasse conjectured that the norm and the Davenport–Hasse relations were the only multiplicative relations connecting Gauss sums over Fp. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture. This counterexample was a new type of multiplicative relation, called a sign ambiguity, involving a ± sign not connected to elementary properties of Gauss sums. In this paper, we give an explicit product formula involving Gauss sums which generates an infinite class of new sign ambiguities, and we resolve the ambiguous sign by using Stickelbergerʼs theorem