## Genus numbers of cyclic and dihedral extensions of prime degree

### Volume 192 / 2020

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

#### Abstract

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 $O(X^{5/8+\epsilon })$.