Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $\mathbb{Z}_p$ of $p$-adic integers.

Let $M$ be the compositum (inside some fixed algebraic closure) of all unramified Galois abelian extensions of $L$ whose degrees over $L$ are powers of $p$. Put $X = \mathrm{Gal}(M/L)$.

Iwasawa theory studies (among many other things) the structure of $X$ as a $\mathbb{Z}_p [[ \Gamma ]]$-module, thus obtaining information about the $p$-Sylow subgroup of the ideal class group of finite subextensions of $L/K$.

Consider the following variant. Let $M_0$ be the compositum of all unramified Galois abelian extensions of $L$ whose degrees over $L$ are $p$. Put $X_0 = \mathrm{Gal}(M_0/L)$.

Did someone consider the structure of $X_0$ as an $\mathbb{F}_p[[\Gamma]]$-module?

Somewhat differently:

What are some results about the $p$-torsion (or $p$-rank) of $\mathcal{Cl}(K_n)$ where $K_n$ is the unique extension of $K$ in $L$ with $[K_n :K] = p^n$ ?

Has some formula (similar to the classical one by Iwasawa with $\lambda$ and $\mu$) been worked out?

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    $\begingroup$ $X_0$ is $X/pX$. Iwasawa's theorem on the growth of the $p$-primary part of the class group tells you more than the size, it also tells us about its structure. $\endgroup$ – Chris Wuthrich Apr 29 '18 at 20:31
  • $\begingroup$ @ChrisWuthrich, you are right, but can you please elaborate? $\endgroup$ – Pablo Apr 29 '18 at 20:33
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    $\begingroup$ For instance: The $p$-rank of the class group of $K_n$ is bounded if and only if $\mu=0$ (Washington Prop 13.23). In that chapter, you will find the tools to answer the second question. $\endgroup$ – Chris Wuthrich Apr 30 '18 at 9:17
  • $\begingroup$ @ChrisWuthrich Thanks! Is there an example where the $p$-rank of the class group of $K_n$ grows as a linear function of $[K_n : K]$ ? $\endgroup$ – Pablo Apr 30 '18 at 9:21
  • $\begingroup$ Isn't it just $\mu\cdot n + \mathbf{O}(1)$ ? If so you need to stay away from the cyclotomic one as we expect $\mu=0$ there. $\endgroup$ – Chris Wuthrich Apr 30 '18 at 14:48

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