Let p be a prime integer and m be an integer, not divisible by p. Let K be the splitting field of XK!1 over the prime field % N . Solving the Gauss sums problem of order m in characteristic p means determining Gauss sums of all multiplicative characters of K of order dividing m. Our aim is to solve this problem when the subgroup 1p2 is of index 2 in (9/m9)*.1998 Academic Press
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