= Nonprincipal cyclotomic fields of conductor a prime power
{title2=$3\mid h_{\mathbb Q(\zeta_{59^r})}\quad(r\geq1)$}
Every field $\mathbb Q(\zeta_{59^r})$ contains $\mathbb Q(\sqrt{-59})$ and is totally ramified over it at the prime above $59$. The <class number divisibility under total ramification> therefore shows that its <class number> is divisible by three. Its ring of integers is not a <principal ideal domain>. The degrees $58\cdot59^{r-1}$ strictly increase, giving infinitely many distinct examples.
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