Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then
where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields.
Consider the family \( {\mathcal{K}}_n \), where K ∈ \( {\mathcal{K}}_n \) if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ \( {\mathcal{K}}_n \) with
where the implied constant depends only on n.
This somewhat improves the analogous result h ≫ d 1/2/(log d)n of W. Duke [MR 1966783 (2004g:11103)].
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 429, 2014, pp. 193–201.
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Fomenko, O.M. On the Class Numbers of Algebraic Number Fields. J Math Sci 207, 934–939 (2015). https://doi.org/10.1007/s10958-015-2416-3
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DOI: https://doi.org/10.1007/s10958-015-2416-3