An MSc student asked me if I knew an example of a prime $p$ and some finite layer $K\_n$ in the cyclotomic $\mathbf{Z}\_p$-extension of $\mathbf{Q}$ (so $[K\_n:\mathbf{Q}]=p^n$) which had non-trivial class group. My gut feeling was that fixing a $p$ and then going up the tower was a bad idea in the sense that going along might find a counterexample quicker, so I tried going along instead, but my computer is having trouble looking at the bottom level when $p$ starts getting bigger than about 30. So now I'm just confused. Presumably this question has been raised before? Can anyone enlighten me with a counterexample or reassurance that this is a standard open question?



Non-trivial class number at some finite level in the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$?
I found the notes of Coates' seminar I alluded to in my comment above. He said the following: 


For $n \ge 1$, let $h(n)$ denote the class number of the unique cyclic degree $n$ extension contained in the compositum of all the cyclotomic $\mathbb{Z}\_p$-extensions for $p \mid n$. It is apparently not too difficult to show that if $n \mid m$, then $h(n) \mid h(m)$, and that if $n$ is a power of $p$, then $h(n)$ is prime to $p$.


Then Weber (not Kronecker, but close!) has conjectured that $h(2^n) = 1$ for all $n$, and this is known for $n \le 5$ and for $n=6$ conditional on GRH. More generally, it is a folklore conjecture that $h(n) = 1$ for all prime power values of $n$.


Horie, Fukuda and Komatsu have recently shown that $h(62)$ is divisible by 31, the first known example of an integer $n$ where $h(n) > 1$.


So the verdict seems to be that your student's question is a well-known open problem.


