A class number formula for the p-cyclotomic field

Let p be an odd prime number and G = Gal(Q(zeta p)/Q). Let S-G be the classical Stickelberger ideal of the group ring Z[G]. Iwasawa [6] proved that the index [Z[G](-):S-G(-)] equals the relative class number h(p)(-) of Q( p). In [2], [4] we defined for each subgroup H of G a Stickelberger ideal S-H Of Z[H], and studied some of its properties. In this note, we prove that when p equivalent to 3 mod 4 and [G:H] = 2, the index [Z[H]:S-H] equals the quotient h(p)(-)/h(Q(root-p)).