On the Stickelberger ideal and circular units of a compositum of quadratic fields. (English) Zbl 0840.11044

Kučera, Radan

J. Number Theory 56, No. 1, 139-166 (1996).

Let \(k\) be a compositum of quadratic number fields such that its genus class field in the narrow sense (i.e. unramified at the finite primes) does not contain \(\sqrt{- 1}\). For such fields the author defines a group \(C\) of circular units (for precise definitions, see the article) containing the group \(C'\) introduced by W. Sinnott [Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)], and computes its index \((E: C)\) in the full group \(E\) of units in \(k\). This index \((E: C)\) is the product of an explicitly given power of 2, Hasse’s unit index \(Q\), and the class number \(h\) of the maximal real subfield of \(k\). After comparing \(C\) to Sinnott’s group \(C'\) by computing the index \((C: C')\) (this turns out to be a power of 2 bounded by an expression involving the number of ramified primes), the author uses his results to prove that \(h\) is divisible by a certain power of 2 depending essentially on the degree \((k: Q)\) and the number of ramified primes.
Similar class number factors have been given previously for full cyclotomic number fields by D. Kubert [J. Reine Angew. Math. 369, 192-218 (1986; Zbl 0584.12003)] using similar methods, and by G. Cornell [Bull. Am. Math. Soc., New Ser. 8, 55-58 (1983; Zbl 0519.12004)] using relative genus theory (see also the remarks of G. Cornell and M. Rosen [Compos. Math. 53, 133-141 (1984; Zbl 0551.12006)] in this connection).
The author then studies the Stickelberger ideal of \(k\); this allows him to compute an index of ideals in the group ring \(Z[\text{Gal}(k/Q)]\) introduced in Sinnott’s paper (loc. cit.).
The results of this paper have been used in the author’s articles, Acta Arith. 67, No. 2, 123-140 (1994; Zbl 0807.11050) and J. Number Theory 52, No. 1, 43-52 (1995).

Reviewer: F.Lemmermeyer (Heidelberg)

MSC:

11R27	Units and factorization
11R11	Quadratic extensions
11R29	Class numbers, class groups, discriminants

Keywords:

quadratic number fields; circular units; class number; Stickelberger ideal

Citations:

Zbl 0465.12001; Zbl 0584.12003; Zbl 0519.12004; Zbl 0551.12006; Zbl 0807.11050