nt.number theory - Kummer generator for the Ribet extension

Let me first add that Herbrand wasn't the first to publish his result; it was obtained (but with a less clear exposition) by Pollaczek (Über die irregulären Kreiskörper der $ell$-ten und $ell^2$-ten Einheitswurzeln, Math. Z. 21 (1924), 1--38).

Next the claim that the class field is generated by a unit is true if $p$ does not divide the class number of the real subfield, that is, if Vandiver's conjecture holds for the prime $p$.

Proof. (Takagi)
Let $K = {mathbb Q}(zeta_p)$, and assume that the class number of
its maximal real subfield $K^+$ is not divisible by $p$. Then any
unramified cyclic extension $L/K$ of degree $p$ can be written in
the form $L = K(sqrt[p]{u})$ for some unit $u$ in $O_K^times$.

In fact, we have $L = K(sqrt[p]{alpha})$ for some element
$alpha in O_K$. By a result of Madden and Velez, $L/K^+$ is normal
(this can easily be seen directly). If it were abelian, the subextension
$F/K^+$ of degree $p$ inside $L/K^+$ would be an unramified cyclic
extension of $K^+$, which contradicts our assumption that its class
number $h^+$ is not divisible by $p$.

Thus $L/K^+$ is dihedral. Kummer theory demands that
$alpha /alpha' = beta^p$ for some $beta in K^+$, where
$alpha'$ denotes the complex conjugate of $alpha$.

Since $L/K$ is unramified, we must have $(alpha) = {mathfrak A}^p$.
Thus $(alpha alpha') = {mathfrak a}^p$, and since $p$ does not
divide $h^+$, we must have ${mathfrak a} = (gamma)$, hence
$alpha alpha' = ugamma^p$ for some real unit $u$.

Putting everything together we get $alpha^2 = u(betagamma)^p$,
which implies $L = K(sqrt[p]{u})$.

If $p$ divides the plus class number $h^+$, I cannot exclude the possibility that the Kummer generator is an element that is a $p$-th ideal power, and I cannot see how this should follow from Kummer theory, with or without Herbrand-Ribet.

If $p$ satisfies the Vandiver conjecture, the unit in question can be given explicitly, and was given explicitly already by Kummer for $p = 37$ and by Herbrand for general irregular primes satisfying Vandiver: let $g$ denote a primitive root modulo $p$, and let $sigma_a: zeta to zeta^a$. Then
$$ u = eta_nu = prod_{a=1}^{p-1} bigg(zeta^frac{1-g}{2}
frac{1-zeta^g}{1-zeta}bigg)^{a^nu sigma_a^{-1}}, $$
where $nu$ is determined by $p mid B_{p-nu}$.

Here is a survey on class field towers based on my (unpublished) thesis on the explicit construction of Hilbert class fields that I have not really updated for quite some time. Section 2.6 contains the answer to your question for primes satisfying Vandiver.