  Let $s_d(p,a) = \min \{k | a = \sum_{i=1}^{k}a_i^d, a_i\in \ff_p^*\}$ be the
smallest number of d-th powers in the finite field F_p, sufficient to represent
the number a in F_p^*. Then $$g_d(p) = max_{a in F_p^*} s_d(p,a)$$ gives an
answer to Waring's Problem mod p.
  We first introduce cyclotomic integers n(k,\nu), which then allow to state
and solve Waring's problem mod p in terms of only the cyclotomic numbers (i,j)
of order d.
  We generalize the reciprocal of the Gaussian period equation G(T) to a
C-differentiable function I(T) in Q[[T]], which also satisfies I'(T)/I(T) in
Z[[T]]. We show that and why a\equiv -1 mod F_p^{*d} (the classical "Stufe", if
d = 2) behaves special: Here (and only here) I(T) is in fact a polynomial from
Z[T], the reciprocal of the period polynomial.
  We finish with explicit calculations of g_d(p) for the cases d = 3 and d = 4,
all primes p, using the known cyclotomic numbers compiled by Dickson.
