Quadratic Gauss Sums

Letpbe a prime integer andmbe an integer, not divisible byp. LetKbe the splitting field ofXm−1 over the prime field p. Solving the Gauss sums problem of ordermin characteristicpmeans determining Gauss sums of all multiplicative characters ofKof order dividingm. Our aim is to solve this problem when the subgroup p is of index 2 in ( /m )*.