Question: Are the class numbers of $\mathbb{Q}(\cos(\frac{2\pi}m))$ $O(m^n)$ for some fixed $n$?

Remark: If such $n$ exists, then $n\geq2$. By the paper unit groups and class numbers of real cyclic octic fields by Yuan-Yuan Shen, there exists infinitely many cyclic octic fields with conductor $32b$ and class numbers at least $c(\epsilon)b^{3-\epsilon}$ for every $\epsilon>0$. As every number field extension $E/F$ satisfies $h(F)$ divides $[E:F]h(E)$, the class numbers of such $\mathbb{Q}(\cos(\frac{2\pi}{32b}))$s are at least $c(\epsilon)b^{2-\epsilon}$.

A key feature of Yuan-Yuan Shen's octic fields is that they have regulators of size $\log^6(b)$. If such an infinite family of number fields (i.e. real abelian with poly-logarithmic regulators) exists for every fixed degree, the answer for this question will be "no".