×

The nonquadratic imaginary cyclic fields of \(2\)-power degrees with class numbers equal to their genus class numbers. (English) Zbl 0919.11071

In a previous paper, the author proved that there are only finitely many imaginary abelian number fields such that their class numbers are equal to their genus class numbers, and that except for quadratic and bicyclic quadratic fields an upper bound on their conductors can be effectively found [cf. S. Louboutin, Manuscr. Math. 91, 343-352 (1996; Zbl 0869.11089)]. In this paper he intends to determine all the imaginary cyclic fields of 2-power degrees with class numbers equal to their genus class numbers.
For this purpose he computes lower bounds on relative class numbers of imaginary cyclic fields of 2-power degrees \(\geq 4\), and upper bounds on the conductors of these fields with class numbers equal to their genus class numbers. Thus he shows that there are 28 such imaginary cyclic number fields, namely 23 quartic fields, 4 octic fields and one field of degree 16, i.e. the cyclotomic field \(\mathbb{Q}(\zeta_{17})\) and lists them explicitly in tables.

MSC:

11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
11R42 Zeta functions and \(L\)-functions of number fields

Citations:

Zbl 0869.11089
PDFBibTeX XMLCite
Full Text: DOI