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For imaginary quadratic number fields $K$ of fundamental discriminant $-D$, the Brauer-Siegel theorem implies that the class number $h(D)$ of $K$ is "close" to $\sqrt{D}$, more precisely for any $c\in(0,1/2)$, there exists strictly positive constants $A_c$ and $B_c$ such that $$ A_cD^{\frac{1}{2}-c}\leq h(D) \leq B_cD^{\frac{1}{2}+c} $$

I'd like to know if similar results are known for more general fields, typically quadratic CM extensions of totally real fields. It seems that in the more general case additional inputs have to be included, such as regulators. Can one simply reduces the asymptotic case to some rational functions in discriminants?

Thanks a lot!

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Yes. See Stark's paper "Some effective cases of the Brauer-Siegel theorem", and the papers "On Artin's conjecture and the class number of certain CM fields I, II" of Hoffstein-Jochnowitz. – David Hansen Feb 21 '11 at 13:20

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