Ribet's paper on the Herbrand-Ribet theorem constructs a representation $\rho: Gal(\overline{\Bbb Q}/\Bbb Q) \to GL_2(\mathbb F_q)$ where $q = p^r$ of the specific form: $ \begin{bmatrix} 1 & *\\ 0 & \chi \end{bmatrix}$ where $\chi$ is a power of the cyclotomic character mod $p$.

In particular, if we let $K$ be the kernel of $\chi$, the matrix is of the form $ \begin{bmatrix} 1 & *\\ 0 & 1 \end{bmatrix}$ and our representation looks like $\rho: Gal(\overline{\Bbb Q}/K) \to \mathbb F_q$.

The image is a subgroup (under addition) of $\mathbb F_q$ and if $B$ is the field fixed by the kernel of this map, then Ribet shows that $B/K$ is an unramified extension with abelian Galois group of the form $(p,\dots,p)$ on which $Gal(K/\Bbb Q)$ acts by $\chi^{-1}$.

I believe it is conjectured that $B/K$ should be a cyclic extension. Morever, by the main conjecture of Iwasawa theory, we can deduce that the degree of $B/K$ is less than the p-adic valuation of a particular Bernoulli number and most of the time, this is just $1$.

This suggests that $B/K$ should always be cyclic - can we prove this directly?