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Genus numbers of cyclic and dihedral extensions of prime degree

Volume 192 / 2020

Henry H. Kim Acta Arithmetica 192 (2020), 289-300 MSC: Primary 11R21; Secondary 11R29. DOI: 10.4064/aa181213-18-6 Published online: 8 November 2019


We study genus numbers of cyclic and dihedral number fields of prime degree $l\geq 5$. For cyclic number fields, we obtain definitive results. For the dihedral case, by assuming a conjecture on the average of $l$-part class numbers, we obtain partial results on the number of dihedral number fields, and their genus numbers. In particular, the number of $D_5$-extensions of discriminant $\leq X$ whose associated quadratic extensions are imaginary is $O(X^{5/8+\epsilon })$.


  • Henry H. KimDepartment of Mathematics
    University of Toronto
    Toronto, ON M5S 2E4, Canada
    Korea Institute for Advanced Study
    Seoul, Korea

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