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Stickelberger ideal of a compositum of a real bicyclic field and a quadratic imaginary field. (English) Zbl 1141.11052
Let \(K\) be an abelian field of degree of \(l^2\) with \(G= \text{Gal}(K/ \mathbb Q)\cong \mathbb F\times \mathbb F\), \(l\) being an odd prime not ramifying in \(K\). Let \(F\) be a quadratic imaginary field such that all primes ramifying in \(K\) split in \(F\). Then the index of the Stickelberger ideal of the compositum \(KF\) is
\[ [A{:}S]=\frac 1Q \cdot l^{\frac 12 (l-1)l-\sum \limits _{i<a_i} (a_i-i)}\cdot h^-_{KF}. \] Here if \(f=p_1p_2\cdots p_s\) is the conductor of \(K\), \(K_i\) the non-trivial proper subfields of \(K\) and \(P_i\) the index set of all primes ramifying in \(K\) but not in \(K_i\), the numbers \(a_i\) are defined as follows: \[ a_i= \begin{cases} -\infty &\qquad \text{if there is a} \,j\in P_i\, \text{such that}\, p_j \,\text{splits in}\, K_i,\\ l-1-\mid P_i\mid &\qquad \text{if there is no such}\, j\in P_i. \end{cases} \]
We order the fields \(K_i\) so that \(a_0\geq a_1\geq \cdots \geq a_l\). For the Sinnot’s index we have \[ (e^-R\:e^-U)=l^{\frac 12 (l-1)l-\sum \limits _{i<a_i} (a_i-i)}. \]
MSC:
11R20 Other abelian and metabelian extensions
11R29 Class numbers, class groups, discriminants
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References:
[1] KRAEMER P.: Circular units in a bicyclic field. J. Number Theory 105 (2004), 302-321. · Zbl 1046.11078
[2] KUČERA R.: On the Stickelberger ideal and circular units of a compositum of quadratic fields. J. Number Theory 56 (1996), 139-166. · Zbl 0840.11044
[3] SINNOTT W.: On the Stickelberger ideal and the circular units of an abehan field. Invent. Math. 62 (1980), 181-234. · Zbl 0465.12001
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