Cyclotomic constructions of skew Hadamard difference sets

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of F q of order N = 2 p 1 m , where p 1 is a prime and m a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.