An algorithm for computing the Stickelberger ideal for multiquadratic number fields

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.