On the parity of the class number of the field of q-th roots of unity. (English) Zbl 0703.11052

Estes, Dennis R.

Rocky Mt. J. Math. 19, No. 3, 675-682 (1989).

Kummer began investigations of the parity of the class number \(h_ q\) of the cyclotomic field, \({\mathbb{Q}}(\zeta_ q)\) of q-th roots of unity over the rationals where q is prime. Hasse refined Kummer’s results for imaginary cyclic extensions of \({\mathbb{Q}}\). The author cites many references concerning work on the parity of the class numbers of Abelian fields (including that of the reviewer) as motivation for his research in this paper wherein he proves: If q and \(p=(q-1)/2\) are primes with 2 being inert in \({\mathbb{Q}}(\zeta_ p+\zeta_ q^{-1})\) then \(h_ q\) is odd.

Reviewer: R.A.Mollin

MSC:

11R18	Cyclotomic extensions
11R29	Class numbers, class groups, discriminants

Keywords:

parity; class number; cyclotomic field