A GENERALIZATION OF A LEMMA OF SULLIVAN

Consider an irreducible polynomial of the form f(X) = X-p - aX - b is an element of F[X] and alpha a root of f(X), where F is a field of characteristic p. In 1975, F.J. Sullivan stated a lemma that provides the trace, taken with respect to the extension F(alpha)/F, of elements of the form alpha(n), where 0 <= n <= p(2) - 1. We present a generalization of Sullivan's Lemma and provide another proof of the original lemma. We explain how computing Tr(alpha(n)) for n < p(r) can be reduced to computing the traces Tr(alpha(m)) for all m <= r(p - 1).