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A Class Number Problem in the Cyclotomic $\mathbf{Z}_3$-extension of $\mathbb{Q}$
Let $\Omega_n$ be the $n$-th layer of the cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}$ and $h_n$ the class number of $\Omega_n$. We claim that if $\ell$ is a prime number less than $10^4$, then $\ell$ does not divide $h_n$ for any positive integer $n$.