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

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).

MSC:

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