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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