Gauss sums and binomial coefficients

Suppose p = tn + r is a prime and splits as p(1)p(2) in Q(root-t). Let q = p(f) where f is the order of r modulo t, chi = omega((q-1)/t) where omega is the Teichmuller character on F-q, and g(chi) is the Gauss sum. For suitable tau(1) is an element of Ga1(Q(zeta(l), zeta(p))/Q) (i = 1,...,g), we show that Pi(i=1)(g) tau(l)(g(chi)) = p(alpha)((a+b root-t/2) such that 4p(h) = a(2) + tb(2) for some integers a and b where h is the class number of Q(root-t). We explicitly compute a mod (t/gcd(8, t)) and a mod p, in particular, a is congruent to a product of binomial coefficients modulo p. (C) 2002 Elsevier Science (USA).