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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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Lower bounds are $h^+ \ge 1$ and $h^- \ge (2\pi)^{-p/2}p^{(p-25)/4}$ where the latter holds for $p > 200$ (can be found in Lang, Cyclotomic fields). – Ralph Jan 29 '12 at 23:21
Great, thank you. – Jonah Jan 29 '12 at 23:38
Related to the original question, are there bounds for the prime-to-$p$ part of $h^-$? – Jonah Feb 4 '12 at 22:24
I would be interested by the same question for imaginary quadratic fields. Is there anything known? – Bernikov Feb 8 '12 at 12:51
@Bernikov: There's a lot to be said on that front. A good starting point is Goldfeld's "THE GAUSS CLASS NUMBER PROBLEM FOR IMAGINARY QUADRATIC FIELDS". – Cam McLeman Feb 8 '12 at 13:55

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