11), a real abelian field F satisfies ([H.sub.p]) if and only if F = Q([square root of 5]) or Q([square root of (13)]) (resp.

Denote by [h.sup.M] the class number of a number field M, and by [h.sup.-.sub.M] the relative class number when M is an imaginary abelian field. We set [h.sup.-.sub.p] = [h.sup.-.sub.M] when M = Q([[zeta].sub.p]).

At present, we have no example of an abelian field F which satisfies ([H.sub.p]) for some p but [h.sub.F] = > 1.

Let F be a real abelian field satisfying ([H.sub.p]), and N = F([square root of (-p)]), K = F([[zeta].sub.p]).

Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent.

In particular, all the abelian fields F satisfying ([H.sub.3]) are determined in Carter [3] and the author [10] when [F : Q] = 2, and by Yoshimura [20] when [F : Q] > 2.

The main purpose of this note is to deal with real abelian fields satisfying ([H.sub.p]) for those odd prime numbers p with h(Q([square root of (-p)])) = 1, where h(Q([square root of (-p)])) is the class number of Q([square root of (-p)]).

From Proposition 1 and [5, Theorem 1.1] mentioned above, we obtain the following assertion using some computational results on abelian fields.