Let $K=\mathbb{Q}(\mu_p)$ with class number $h=h^+h^$, where as usual $h^+$ is the class number of the maximal real subfield of $K$. My question is whether there is an effective lower bound for $h$ (which I imagine would be given through one for $h^)$. I've seen upper bounds for $h^$ (e.g. here), as well as an asymptotic formula (e.g. here).
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