TITLE: A question on Iwasawa theory
QUESTION [5 upvotes]: Let $p$ be an odd prime. Let $K_{\infty}$ be the field extension of $\mathbb{Q}$ generated by all $p^n$-th roots of unity. Let $M/K_{\infty}$ be the maximal abelian $p$-extension of $K_{\infty}$ unramified outside $p$. Denote its Galois group by $\mathcal{X}$. Then the  Iwasawa main conjecture concerns the structure of $\mathcal{X}^{+}$ (here, $\pm$ means the $\pm1$-eigenspace of complex conjugation).  My question is whether one can  say anything about the structure of $\mathcal{X}^-$? 
I don't find the answer in the basic textbook (e.g. Lang's Cyclotomic fields & Washington's Introduction to cyclotomic fields). I wonder whether $\mathcal{X}^-$ is a free $\mathbb{Z}p[[\Gamma]]$-module where $\Gamma$ is the Glois group of $K_{\infty}/\mathbb{Q}(\zeta_{p})$?
Can somebody tell me the answer?
Thank you very much.

REPLY [2 votes]: Thank you very much.
Kummer-Vandiver conjecture will imply $X^+=0$, not just $X^+\sim 0$. Recently, I found some connection between Kummer-Vandiver conjecture and the freeness of $\mathcal{X}^-$ as a $\mathbb{Z}p[[\Gamma]]$-module. That is why I asked about $\mathcal{X}^-$.