Geodesic
An algorithm for computing the Stickelberger ideal for multiquadratic number fields
Prikladnaâ diskretnaâ matematika, no. 1 (2021), pp. 9-30

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We present an algorithm for computing the Stickelberger ideal for multiquadratic fields $K=\mathbb{Q}(\sqrt{d_1}, \sqrt{d_2},\ldots,\sqrt{d_n})$, where the integers $d_i \equiv 1 \bmod 4$ for $i \in \{1, \ldots, n\} $ or $d_j \equiv 2 \bmod 8$ for one $j \in \{1, \ldots, n \}$; all $d_i$'s are pairwise co-prime and square-free. Our result is based on the paper of Kučera [J. Number Theory, no. 56, 1996]. The algorithm we present works in time $\mathcal{O}(\lg \Delta_K \cdot 2^n \cdot \mathrm{poly}(n) )$, where $\Delta_K$ is the discriminant of $K$. As an interesting application, we show a connection between Stickelberger ideal and the class number of a multiquadratic field.
Keywords: multiquadratic number field, Stickelberger element, Stickelberger ideal, class group of multiquadratic field. 
@article{PDM_2021_1_a1,
     author = {E. A. Kirshanova and E. S. Malygina and S. A. Novoselov and D. O. Olefirenko},
     title = {An algorithm for computing the {Stickelberger} ideal for multiquadratic number fields},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {9--30},
     publisher = {mathdoc},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2021_1_a1/}
}
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E. A. Kirshanova; E. S. Malygina; S. A. Novoselov; D. O. Olefirenko. An algorithm for computing the Stickelberger ideal for multiquadratic number fields. Prikladnaâ diskretnaâ matematika, no. 1 (2021), pp. 9-30. http://geodesic.mathdoc.fr/item/PDM_2021_1_a1/